I actually think I can explain Trump's popularity, but it boils down to a "lesser evil" theory. Aside from any personal failings other candidates may have individually or collectively, there's a very anti-establishment mood and--say what you want about Trump--he's about as anti-establishment as you can get. By most measures, the last two administrations have produced the two worst economic recoveries since the Great Depression, and especially following the dot com boom and cold war peace dividend of the 90s they've been very bitter pills indeed which has helped quite a few folks decide the status quo isn't a good idea.
Something which I believe voters in parliamentary systems don't really grasp is all the implications of the two-party system. You guys can decide that, say, environmentalism is the most important issue for you, vote accordingly, and know that your representatives will work themselves into a coalition as they think best. In the U.S., we do have "single-issue voters" but our districts create more of a "winner take all" situation with votes than you often see in a parliamentary system, and we often have to swallow views wildly at variance of our own on all sorts of social, economic, or whatever based on how we prioritize things individually. Our parties mean our coalitions are basically pre-packaged, which really decreases the power of any contingent outside the dominant one.
Also, something else which is fairly unique about the American system is that our government is actually designed to do as little as possible. Most of you have Prime Ministers there to execute the will of the legislative branch, we have a system of "checks and balances" designed to keep them playing off against each other, with the Supreme Court as another player to put the brakes on both of them. Gridlock isn't a bug, it's a design feature.
We've got some other problems, like the same states always leading the primaries, and I think the role of money in elections is overblown (the Chamber of Commerce routinely proves it's definitely huge, while Jeb Bush has spent most of the last year proving you can be utterly irrelevant despite having all the money), but I think those are the issues most alien to members of other democracies/republics.