To fatten someone is to feed them until they become bigger and fatter. Your grandmother might see a tiny baby and say, "His mother needs to fatten him up!"
A farmer might spend weeks trying to fatten his prize pigs before he sells them for meat, and you may visit relatives who keep cooking you delicious food in an attempt to fatten you up. Figuratively, you can also fatten things like your bank account, by making it larger. Before the 1550s, the verb form of this word was fat — and both fat and fatten come from the Old English word fætt, "fat or plump."