Why is acetic anhydride used in aspirin synthesis?
Because acetyl salicylic acid, aka Aspirin, has an acetate group, and acetic anhydride has one that is a good leaving group.
The typical lab-ready reaction for this synthesis is:
Basically, the acid acts as a catalyst (not protonating water though, because acetic anhydride is violently reactive with water), and the hydroxyl on 2-hydroxybenzoic acid acts as a nucleophile. It doesn't work too quickly though, so the heat is needed to speed it up.
The acetic anhydride essentially ends up transferring its acetyl group to 2-hydroxybenzoic acid.
The mechanism is as follows:
You can also tell from the full procedure (and from the name "anhydride", resembling "anhydrous", meaning "lacking water...") that the water easily cleaves acetic anhydride (into acetic acid).
As a result, it's easy to get rid of all the acetic anhydride when you're done with it. It makes cleaning up easier, because although acetic anhydride is actually quite hazardous, dilute acetic acid is pretty harmless, and it's generally easy to find DI water nearby to further dilute the acetic acid if necessary. Ultimately, relatively green chemistry.