You're somewhat confused, which is understandable since intellectual property rights and D&D is a confusing issue.
OGL
The OGL is a specific license with specific terms. D&D 3e/3.5e was made open for others' use under the OGL and the open portion was published as a SRD, or System Reference Document. Other games derived from the d20 SRD (like Pathfinder and Mutants and Masterminds) and totally unrelated games, like FATE, use the OGL. 4e used a non-open license called the GSL, and previous versions had no available user license except under contract with TSR. You can read the blog post Open Gaming for Dummies to understand these very specific legal terms.
There are some "retroclones" that use the OGL and actually pull the OGL 3.5e information from the d20 SRD into something that looks more like an older version of D&D. See also What content can I reproduce from Pathfinder? that has a similar discussion specific to Pathfinder. Some people use the OGL to put out modules for various other non-OGL D&D versions, but that's actually somewhat complex and you need to understand the rest of what's going on in IP land to know how and why.
Wizards has also released 5e - or at least some of it - under the OGL as well. You can download it and the SRD on their site. Be careful, the SRD only has a subset of the game's content, and it's all that can be used under the OGL.
Intellectual Property
Let's talk about using game IP in other games outside the scope of the OGL. The more general discussion in Can I use existing game mechanics in my own designs? has a lot of relevant details for you here, as well as Is it legal to "use" (reference) copyrighted material in a way that requires ownership of its original publication? You have three main areas of IP concern to contend with - copyright, trademark (including trade dress) and patents.
Copyright Basics
First of all, "fair use" does not apply in this case. At all. It's one of those phrases people like to use on the Internet but don't understand. If you up and use text from another work in your work, and it's not part of the very limited scope of fair use (educational, review, journalistic, etc.) it's illegal.
Secondly, game mechanics cannot be copyrighted. "Copyright does not protect the idea for a game, its name or title, or the method or methods for playing it. Nor does copyright protect any idea, system, method, device, or trademark material involved in developing, merchandising, or playing a game." (United States Copyright Office, "Copyright Registration of Games").
However, "the text matter describing the rules of the game" may be protected by copyright "if it contains a sufficient amount of literary or pictorial expression". For example, a passage describing character creation for the Clans of the Sun and Moon, explaining their society and why they tend to have the skills they do, would probably be protected by copyright. A section merely describing the steps involved in rolling a number of dice or expending a number of points on X attributes would probably not be. (rpglibrary.org) Note that this means even charts and tables can be copyrighted. Everything's copyrighted, there's not a registry or list or anything (well there is, but stuff not on it is still copyrighted).
In general, this is the escape clause some people use to put out supplements/modules that simply have "goblins" and "skeletons" in them which may have "HD: 2+2" or "a DC 12 Perception check" outside the scope of the OGL. If it's "mechanics" and not "expressive" that's legal. But head onward and read about trademark...
Trademark
However, various aspects of the game rules may also be trademarked. Several D&D monsters are, and in general things that would usually be Product Identity in an OGL game can be. Characters, game worlds, etc. Without a more specific license allowing it, you absolutely cannot set your module in the Forgotten Realms and have Elminster fighting a beholder in it. The reason many of the retroclones do use the OGL is so they don't have to fret over whether somewhat unique terms like "Hit Dice" end up getting trademarked or not, since as a term you're explicitly allowed to use it if you are using the OGL.
You will note that many products/publishers, out of fear of trademark won't say they are "compatible with Dungeons & Dragons Fifth Edition" but instead say circuitous things like "for 5e!" Normally, expressing compatibility with something is an OK use of trademark, Hasbro themselves lost a lawsuit on that v. RADGames about a company claiming Monopoly compatibility for an add-on pack. But the OGL giveth and the OGL taketh away, the OGL says you can't claim compatibility in its terms so those using it can't legally say "D&D".
Note that trade dress also means you can't use graphics/art/logos/etc that make your product look "too much like" products from an existing publisher.
You can do trademark searches to look for trademarks, see the discussion in Are the names of the more generic planes copyrighted under the OGL as Product Identity?
Patent
Game mechanics can be patented, since processes can be patented. I do not believe there is a patent on file for RPGs or D&D but I'm not an IP lawyer. You can do patent searches as well. At least one RPG publisher applied for a RPG patent, check it out here (article on it). Just wanted to mention this for completeness.
Dungeon Masters Guild
Speaking of more specific licenses, Wizards has come out with one! Their "Dungeon Masters Guild" program is designed to specifically allow publishing of 5e material with a number of restrictions, primarily that such content may only be distributed through their online storefront and they get a 50% cut of the sales (though you can make it free - although you can't distribute it other than through their site). The upsides are that (a) you can use the Forgotten Realms as a setting and (b) you can use all of5e rules/content, not just the subset provided in the SRD.
Bottom Line for Modules
"I am not an IP lawyer and this is not legal advice blah blah" but in general you have three paths open to you. All require a nontrivial amount of legal understanding, so taking a RPG.SE answer and going forth to publish would be the height of folly.
But the first is to use the OGL, limit yourself only to things found in the SRD, and don't claim D&D compatibility. Most folks do this, for example Frog God as you note. For an adventure, with a little oversight, you are probably fine with just writing general scenario content with pure rules references.
Or, you can not use the OGL, navigate the copyright/trademark waters yourself (and/or hope Hasbro doesn't care), and even claim compatibility. KenzerCo did this with some products; their owner, David Kenzer, is a lawyer so he knew what he was doing and figured he could hold his own. Since modules usually don't contain large amounts of rules content with the exception of stat blocks, the problem space is pretty narrow.
Now, profit vs it's a freebie on the Web doesn't change most of the legal factors, but it does change immensely the likelihood you'll get in trouble over it, so if you're not really looking to publish for sale, go ahead. If you're starting out and looking to publish for sale, it might be a good idea to do that through an established third party company (like FGG) that can help you navigate these waters.
Or, you can use the Dungeon Masters Guild program, if it suits your needs.
Best Answer
System Reference Documents are for what they say on the tin: they're built as a general reference document for the game system. What goes into them exactly depends on what the authors decided to put in there to build that reference — there's no set standard. Some SRDs are enormously comprehensive libraries of just about everything or even the full text of the game, some are far more limited in coverage.
Primarily, SRDs get targeted at two audiences:
The D&D / d20 SRD is not the only one, so Wikipedia's out of date in talking like it is, though it's certainly the most well known. It existed so third party publishers could legally release a ton of d20 system material, and know what they could legally use. Wizards created the Open Gaming License specifically for releasing the d20 SRD under it.
However, you've come across other SRDs and you're curious about them.
Why do they exist if you could just play the whole game with them?
In the d20 SRD's case, they left out the character creation rules and level progression table so as to hopefully force you to buy a book.
However, for other games, being able to figure out everything is sometimes the point! For these though, it's more a matter of: why would you release a whole game for free!? (I'll talk more about that later.)
The Fate SRD publishes the entirety of Fate Core & Accelerated (plus the System Toolkit, an accessory to them). Those games are both completely free, released under Creative Commons and the OGL and also available in full as PDFs. The SRD site itself was fan-made as a channel for accessing the game in full. If you want to play this game, get yourself the PDF or open the SRD, and go play it and have fun.
Pelgrane Press offers a Gumshoe SRD. Just like Fate Core, it's the full game, free under CC and OGL.
Other times, like in the case of Gregory Woolfe's Traveller SRD, the SRD is just an unofficial fan creation to help wrangle with the system. Probably these shouldn't render buying the book unnecessary when the game's not free — sites that do this tend to receive cease & desist orders if the publisher takes umbrage.
What are the SRD creators getting out of them?
(Some people are just happy releasing things for free and that's an end in itself. However, for this I'm going to talk about the cases assuming someone gets something out of it, and that monetary gain is a longterm goal.)
When made by the owners: the right free stuff means marketing.
So as I mentioned, often this is more about why you'd release a game for free at all. Fred Hicks, cofounder of Evil Hat and creator of Fate, talked about this in his blog. I'm trimming it heavily to the key points, but the whole thing's worth reading if you're interested in this:
Evil Hat gets a large portion of income from Fate being implemented in other games. In a public Evil Hat quarterly sales report, Fred Hicks was surprised Fate Core actually made them money at all:
They released it to get people using the game and get it attention. Making any money directly from providing this material was a bonus — the point is the money that arrives later as a side effect of the free material.
Pelgrane Press is in a similar situation with Gumshoe, which has been extended into Trail of Cthulhu and other investigative games. They don't need to sell Gumshoe itself — if they can get people interested in Gumshoe (because they can actually try it out for free), those people may get interested in other Gumshoe products and buy those, and spread excited word of mouth. This is good!
Worth noting that in Gumshoe's case, the SRD was released as a result of the Hillfolk kickstarter in 2013, when Gumshoe had already been out for several years and sold plenty enough.
In the case of D&D, it was released because Wizard was happy to see other people building their brand and solidifying their status as owning the go-to RPG system (d20 or D&D). This surge in reputation handily translates to money if you can take advantage of it, and Wizards could.
When made by fans: better organisation, and helping other players.
We're social animals. Generally, people like helping people[citation needed]. And sometimes the books we're dealing with are a pain in the arse and we really wish they had a search function or we could link to their pages, and sometimes the people thinking this have the will and means of making it so.
That gives us sites like the Fate and Traveller SRDs, to help us navigate the games we love. (Heck, the Fate SRD is transforming into a fantastic portal for Fate Core, and the author even won a silver ENnie award for their efforts on the site.)
The d20 Pathfinder SRD fan site was also originally just a far more comprehensive version of Paizo's official Pathfinder Reference Document that included all the third party stuff. Nowadays to help pay for hosting, it also operates as a store and advertises that fact all over its pages, so you could say that offering that SRD is now marketing for the store as well.
So, in summary: they contain whatever the author decides you need to be able to reference for a system. They're created and released because someone wanted their brand built or a diehard fan decided their favourite game needed one.