Simple answer: The math says so https://www.youtube.com/watch?v=-4FXLQoLDBA

Longer answer: cardinal systems (the generalized form that approval is part of) is able to capture more information about voter preference. It is easiest to see this with something like range voting. You may be familiar with a question like "Do you agree or disagree with x statement?: strongly disagree, disagree, neutral, agree, strongly agree." That is range. Approval would be "Do you agree or disagree with x statement?: Disagree, agree". Conversely FPTP would present you many statements and you have to pick the one you agree with the most, even though you may agree with many in the list. You can probably see here that you are actually able to capture a fair amount of preference information through this, thus allowing to elect representatives that more accurately represent the population as a whole.

I would frequently vote for more than just one candidate. And people who don't vote because e.g. Biden isn't progressive enough would have someone to vote for who could actually garner a significant number of votes. So maybe turnout improves too. Those other candidates/parties could then get momentum and funding and show up at the debates.

If you did it for just one election, I'd bet something like the president would get the same outcome, but local stuff with half a dozen decent candidates on the ballot would be different. Over time, different candidates winning lower races, plus 3rd party candidates for the big races being able to actually gather momentum would have profound impacts on the top races.

It would even change who shows up on the ballot. Remember in the dem primaries, all the commentary about candidate X drawing votes away from candidate Y, so they had to withdraw from the race to give their policies a better chance of winning? The whole "lanes" thing? Sanders and Warren wouldn't be splitting the progressive vote. And so on. Hell, the dems could nominate biden/harris and another pair in an effort to increase turnout, which would be suicide in FPTP. Or maybe not. I'd expect lots of changes and many would likely be surprising.

You can think about it this way -- approval voting allows you to vote strategically for a candidate (in case you are worried about the lesser of two evils winning), and then allows you to also vote for anyone you actually prefer, guaranteeing this can't hurt the strategically chosen candidate.

This would allow third parties to see real support, and then go on to win subsequent elections.

This exit poll result is one data point among hundreds.

http://scorevoting.net/Maine2014Exit