Some restaurants give a free basket of bread with a meal as a sort of appetizer. Most people either consume all of it or just leave the leftovers.

Ah, OK, my (Chinese) wife is like that, she won't leave anything behind, I don't mind leaving it behind, but I can understand trying to avoid waste since many places will throw away bread which was offered to other customer. And often the bread taste actually good, so it makes sense to take it.

It's to fill stomachs so that the portion sizes seem bigger.

Ah, so it's just that the OP wrote something that was prone to be misparsed.

I read "breadbasket home" as a compound noun and guessed it was some kind of food bank or something, but "home" was really an adverb modifying took. So what was meant was something like "My grandma ... always took [home] the bread from a [restaurant] breadbasket..."

Yeah welcome to split verb-adverb pairs in English!

Many people ask for more bread after they wolf it down and even ask for some to take home.

It might be buttery, lightly seasoned bread or something like rolls, usually fresh-baked.

Ah so like a mild version of this xkcd:

https://xkcd.com/1499/