This is the best explanation I’ve seen so far!

Why Trump?
You’ve been on vacation for two weeks, you come home, and your basement is infested with raccoons. Hundreds of rabid, messy, mean raccoons have overtaken your basement. You want them gone immediately so you hire a guy. A pro. You don’t care if the guy smells, you need those raccoons gone pronto and he’s the guy to do it! You don’t care if the guy swears, you don’t care if he’s an alcoholic, you don’t care how many times he’s been married, you don’t care if he voted for Obama, you don’t care if he has plumber’s crack…you simply want those raccoons gone!You want your problem fixed! He’s the guy. He’s the best. Period.
That’s why Trump. Yes he’s a bit of an ass, yes he’s an egomaniac, but you don’t care. The country is a mess because politicians suck, the Republican Party is two-faced & gutless, and illegals are everywhere. You want it all fixed! You don’t care that Trump is crude, you don’t care that he insults people, you don’t care that he’s friends with Hillary, you don’t care that he changes positions, you don’t care that he’s been married 10 times, you don’t care that he’s never read the bible, you don’t care that he fights with Megyn Kelly, you don’t care that he doesn’t know the difference between Hamas & Hezbolah…this country is weak, bankrupt, & being invaded by illegals and you just want it fixed. And he’s the only guy who seems to understand that.
That’s why Trump. You’re sick of politicians, sick of the Republican Party, and sick of illegals. You just want this thing fixed. Trump may not be a saint, but he’s also not a politician. And he’s not a cowardly Republican. And he says he’ll fix it.
That’s why Trump. You don’t care if the guy has bad hair. You just want those damn raccoons gone. Out of your house. Now.

I don’t necessarily agree with the idea of supporting Trump, but I think this is a good explanation of his appeal. Hopefully, the establishment Republicans will take his popularity as a clue that they need to start listening to the voters. The establishment Democrats need to realize that the popularity of Bernie Sanders is also a reflection of the anger of the voters who feel that they are not being listened to.