The DMG outlines what it means to be a spellcaster:
If the prerequisite is to be a spellcaster, a creature qualifies if it can cast at least one spell using its traits or features, not using a magic item or the like.
The requirement to cast spells from scrolls as referenced by the Spell Scroll magic item (DMG pg. 200) says:
…if the spell is on your class's spell list, you can use [it]…
But creatures don't typically have spells lists as they have traits or features for casting spells.
Does that mean that no creature is able to cast spells from scrolls even those categorized as spell casters? What creatures can cast spells from scrolls?
Note: This question differs from Who can use magic scrolls? as this question pertains to creatures specifically which that question does not address.
Best Answer
Only creatures that use a class's spell list and have spell slots or who have the Ritual Caster feat can use Spell Scrolls.
In the published official material as of 2018, this includes:
And excludes PCs and monsters who do not meet the previous criterion.
You could also rule that having a feature that refers to a class's spell list in its text is sufficient but this leads to very odd scenarios where being granted the choice of picking a spell gives you access to cast spells from spell scrolls, but being given spells directly without the ability to pick one from a spell list does not. If you want to use that interpretation then the following features also let you use spell scrolls:
This ruling reads somewhat differently from the written rules, but it's the only way to reconcile the rules as written with the rules as intended. Let's see how we got there:
Initial RAW reading
As you rightly point out, the requirement to cast spells from scrolls as referenced by the Spell Scroll magic item (DMG pg. 200) is that:
At first this appears straightforward. It means that casting a spell from a Spell Scroll is restricted to creatures that both:
This means that the following classes qualify: Bard, Cleric, Druid, Paladin, Ranger, Sorcerer, Warlock, Wizard.
Conversely, the following classes do not: Barbarian, Fighter, Monk, Rogue
For PCs this simply means having a level in one of the qualifying classes.
What about monsters?
For monsters we can rely on the rules on attunement (DMG pg. 136-137) which states:
This means that only certain spellcasting monsters qualify. Let's first look at the Guardian Naga's Spellcasting trait:
Sure enough a Guardian Naga uses the Cleric spell list and has actual spell slots, so it qualifies as a member of the Cleric class and thus gets to cast spells from the Cleric spell list from Spell Scrolls.
Next lets look at the Innate Spellcasting trait of a Rakshasa:
They make no mention of using a class's spell list and do not have spell slots. Thus they do not qualify as a member of any class and do not get to cast any spells from Spell Scrolls.
What about Arcane Trickster Rogues and the Eldritch Knight?
As stated by Jeremy Crawford and related in the question can an arcane trickster use a spell scroll from the wizard spell list?, Arcane Trickster Rogues and Eldritch Knight Fighters are intended to be able to use Spell Scrolls of the Wizard spell list, which means that they are intended to qualify as members of the Wizard class for the purpose of being eligible to use Spell Scrolls.
How can we justify this? The most straightforward method is to use the same rules we used to determine which monsters qualify as members of a class. Both Arcane Tricksters and Eldritch Knights use the Wizard spell list and have spell slots. This means both sub-classes qualify as a members of the Wizard class and thus gets to cast spells from the Wizard spell list from Spell Scrolls.
What about feats?
No feats grant you any spell slots, but as illustro points out Ritual Caster does say:
The requirement for casting spells from Spell Scrolls is tied to being able to understand what's written on the scroll. If this feat allows you to read from spell scrolls of your chosen class, then that means you must also be able to cast spells from those Spell Scrolls.
This now leaves us with 3 possible interpretation as to why Ritual Caster lets you qualify:
Option B doesn't feel very convincing and option C just feels wrong to me so I will go with option A, but you are of course free to use your favorite interpretation.
And that's about it, we've now built the full list of creatures who can cast spells from Spell Scrolls.
Traits that do not help you qualify
For the sake of completion, here are some examples of traits that do not let a creature qualify to use Spell Scrolls:
Subclasses
Feats
Racial traits