An idiom. Common usage is the equivalent to:
"What do you think of that?!"
or a very defiant "Nyaa, nyaa nyaa nyaa nyaa"
This rhetorical question is particularly useful when spiting someone else, as used in the Good Will Hunting example. "I got her phone number. How 'bout them apples?" Another example might be:
If you're going to complain about the downvoting, I'll nuke your node, instead. How do ya like them apples?
JerboaKolinowski says: Could well be Irish - I spotted it in an Irish newspaper just the other day as a photo caption. Lots of Irish in Boston, right?
Note: while I'm not sure the origin of this phrase, it is not unique to New England. In fact, we here in the MidWest use it all the time.
The best answer is a deadpan "Puréed into applesauce, actually, with a sprinkle of cinnamon. Thanks!"