If someone makes a promise, and they successfully keep their promise even though it was difficult or in doubt, then you can say that that person "came through":
She said that she'd make it to my show, and this time she came through.
Aside from a promise, you can also "come through" on someone's wish or hope. For example, you might talk about a sports team this way:
Man, that was a really tough match, and I didn't know if they'd be able to pull it off, but yeah, they came through!