I feel like this conversation has been had a million times already, but I'll give my shot at it anyway.
People like him because he at least tries to to the right thing from his perspective. For example, he doesn't push Bran because he wants to kill a kid (his first instinct is to save him from falling), he pushes him because the alternative is risking the death of himself, his sister, and all his children; I think most people would do the same in his position (not that most people would get into that position in the first place, but that's besides the point).
His oathbreaking was also an example of that. Which is worse? Breaking your oath to your king, or letting everyone in King's Landing die at the hands of a lunatic, including your father and his army? I think his response to Caitlyn about conflicting oaths is very on the spot, and I also think that breaking his oath in that situation is as close to an inherently good, selfless act as you come in this series; he does even though it will make him hated by all, he will be written into the history books as a honorless traitor, and he'll probably be executed for it (though luckily for him he wasn't).
It's hard to say what Jamie'd do if he was in any position of actual power, but I bet that (if his aversion to responsibility didn't make him neglect his duties, and his lack of guile didn't get him killed) he'd be a better person for that position than most the series' characters, including Daenerys, who is about as intelligent as a log and seems to derive actual pleasure from killing defenceless civilians and her own servants, spurred on as she is by the notion that she's morally superior to those she's killing so that makes it okay for her to murder them. She's happy to kill not for some greater good, but for the pleasure of it; that's evil. She has killed far more people than Jamie too (not even counting the ones her armies have killed in combat), and people call him a bad guy.
The only thing about Jamie's character I haven't liked has been when he killed his cousin, which is, from the perspective of what should be his priorities (family above all), about the only unrepentantly evil thing he's done so far, and which honestly didn't make any sense to me when he did it (and which didn't happen in the books so I honestly don't understand why they made him do it in the TV series, other than just to force him into a 'kick the dog' moment so that people would hate him in spite of his dashing good looks; it was an act extremely incongruent with his character).
If it wasn't for that he's been in love with his sister since he was a kid (the incestuous relationship with which causes all his deplorable behaviour), he'd probably be one of the unambiguous heroes of the story instead of an anti-hero.