As another UK person, it's obvious to me. Basically they've only got a choice between a Democrat and a Republican, and the Democrat candidate, Hillary Clinton, looked "corrupt". And it's almost a worldwide movement - someone who looks different and doesn't come from "the political establishment" looks good. Trump said the right kind of appealing things. Much like Nigel Farage - an outsider who knows how to make the right kind of speeches.
Also bear in mind how American presidential elections work. According to the US constitution, the president is elected by 538 electors chosen by the states. There is a deliberate bias towards small states because that was the only way back in 1787 that the small states amongst the original 13 would agree to the constitution. The constitution also doesn't say how each state has to choose its electors, and it's only since 1868 that all of them have done it by asking the people to vote. (South Carolina was the last to pick its electors without having the people vote.)
And what the states nearly all do is "winner takes all" - whoever wins in the state, even if only by one vote, gets all their electors chosen. So the state will vote 100% for their winner even if they only got 51% of the vote.
Put that together and you see why Trump won. Add up all the votes across the USA and he got fewer votes than Hillary Clinton, but he got more votes in the right places so he won when the electors voted.
This whole system is incredibly controversial. Usually it actually does work out that whoever gets most votes across the nation wins, but in 2016 it didn't.
It's just so totally different from what we do in the UK. They're federal, so states matter however big or small they are, while in the UK we aren't federal. It's a big thing - if you live in the USA, most law that affects you is made by the state legislature and the state governor, it's not USA-wide law, and breaking the law is dealt with by state courts, right up to and including murder. If you kill someone in the USA, whether you could be facing execution and how you can be executed depends on which state you did it in. The president is only really responsible for foreign affairs and defence, and whatever he can do for the USA as a whole. The country really is 50 separate countries that just work together.