I'm not sure about every instance, but every time I have seen a comment from a faithless elector about why they did it, it has been that they did it mostly to illustrate how fragile the electoral college system is.
I think a couple of times there might have also been a bit of a "protest vote" element, where someone said, "I hate our party's candidate, and the election is wrapped up to voice my disapproval this way."
I don't think we'll have any faithless electors in a situation where it would actually change the election outcome.
At least, it has never happened yet in American history, but there's always a first time, eh?