So they've changed it from vote for one to vote for zero. Thanks but it's been a long time since a politician I approved of made it onto my ballot. Better wording would be "Vote for as many of the lesser evils as you want." Otherwise I have to lie in order to vote.
Suppose right now you have a Democrat, a Republican and a moderate. In first past the post, the moderate splits the vote with whichever major party candidate is most like them, causing the one less like them to win. Also, everybody knows this and then only votes for one of the major parties. So then the moderate doesn't bother to run and the only thing on the ballot is the one evil or the other.
With approval voting, if the moderate runs, most of the Republicans approve of the Republican and the moderate (because better the moderate than the Democrat) and most of the Democrats approve of the Democrat and the moderate (because better the moderate than the Republican). So then the moderate runs, and wins. And you get a candidate you might actually want to vote for.