Freekvd's answer covers the basics of gyms, but I think leaves out some important details with regards to Gym mechanics, I have tried to be more thorough in this answer.
Gyms are the only locations (currently) where Pokémon battles can occur. When you are battling at a Pokémon Gym, you are either battling an opposing team, using a full roster of six Pokémon that you have selected, or battling ("training against") your own team, using a single Pokémon that you have selected.
Combat itself is relatively straightforward - single tap repeatedly to perform your Pokémon's basic attack and to charge up your secondary attack, and then long-tap (hold down) to use your secondary attack once it is charged. You can also swipe to the left or right to dodge the enemy's attack, although this is a bit finnicky and the timing can be a bit tricky to get right. Beyond this, the results of a Pokémon battle differ depending on the two types of gyms:
Enemy Gyms
These are gyms that are currently controlled by players who are not on the same team as you (i.e. Instinct (Yellow), Mystic (Blue), or Valor (Red)). When you engage in battle against these gyms, you must pre-select a roster of six Pokémon to fight with. Then you will fight the Pokémon in the gym one by one from weakest to strongest until you have beaten them all (or not). For every enemy Pokémon that you defeat, the prestige of that gym will be reduced. If the prestige is reduced enough, that gym will lose a level and the weakest Pokémon will be kicked out and returned to its trainer. If the gym loses enough prestige and falls below level one, it will turn white (neutral) allowing you to install a Pokémon of your own and claim this gym for your team.
Allied Gyms
These are Gyms that are currently controlled by players who are on the same team as you. When you battle at these gyms, you must select a single Pokémon to fight with, and then combat proceeds as normal against the Pokémon in that gym. Winning these will result in an XP reward, and the prestige of that gym will be increased; however, any Pokémon that you defeat will remain in the gym, unlike battles against enemy gyms. You can also install your own Pokémon in a friendly gym provided there is enough space, which is based on the gym's prestige level (e.g. Prestige 2 gyms can hold two Pokémon). Installing your own Pokémon in a gym can be beneficial because it allows you to collect a reward of Pokécoins and Stardust once every ~20 hours for each Pokémon you have installed in a gym. The maximum any player can have in any singular gym is one Pokémon.
The Silph Road has done quite a bit of research into spawning mechanisms and their findings have been corroborated by much of the community. It isn't necessarily guaranteed to be 100% accurate but the evidence supporting it is overwhelming. In either case, all their work is reproducible and peer-reviewed so it's largely considered to be the most accurate non-official information available.
From what I understand of their findings, the world map is littered with "spawn points" (marked by the rustling grass animations), which spawn one pokemon at regular intervals. Most spawn points trigger once per hour but I've heard of some spawning as often as every half hour (although it is possible that the location simply has two spawn points very close together). A pokemon spawned in this way lasts for 15 minutes before despawning and will be available to any player who is in range during that time. Every spawn location has different timings so one might trigger at :02 past the hour and another might trigger at :29.
I have not yet seen any evidence to suggest that these points may cease spawning at any time for any reason. We do know that they are tied to locations that historically have increased mobile data usage, which is why there are tons in the downtowns of larger cities but very few, if any, out in the country.
Assuming this is all correct, it is theoretically possible to momentarily deplete an area of pokemon by catching everything that is currently spawned but that will only affect you and will only last until the next spawn point triggers.
If I were to speculate, I would say that the "dry area" effect is a description of the 45 minutes per hour that there are no pokemon at a given spawn point. If your area only has one or two spawn points within radar range (especially if they all trigger at roughly the same time), the effect would be highly visible, which is likely what that other user was seeing. I don't know whether the map turning grey is related to this but I would suspect it isn't. Instead, I would chalk that up to a graphical glitch but I don't know that for certain.
Best Answer
There's been a rather good post about how the sightings work on reddit, whereas I will summarize what's written there. Nevertheless, credits go to the OP 'Zakrael'.
For the following steps take this image, which is also contained in above mentioned reddit post.
Keep in mind thought that Pokemon will stay spawned for approx. 15 minutes, so if you take too long, the Pidgey might have despawned and therefore all the work has been useless.