First, when economists talk about profits they mean something entirely different than when accountants (or most normal people) do. An "accounting profit" is when revenue exceeds expenses. An "economic profit" is when you are making money because no one is competing with you.
Thiel is talking about economic profits, and his statement is a tautology. Perfect competition is defined as the state in which no one can make economic profits, and economic profits are defined as the money you make as the result of not being in a state of perfect competition. :)
In terms of a startup, you make your money by inventing something new, or by executing something old better, or by errecting barriers to entry, or exploiting first mover advantage, or by having key patents, or by effective advertising. These are all ways in which we struggle to ensure that there is not perfect competition, in order that we may reap economic profits.
Second, as already mentioned, "perfect competition" is a pretty specifc term, and it has some interesting consequences. It is the state where a all products are perfectly identical. Which in turn means that, oddly, there is none of what we call "competition" in a market with perfect competition. :)
In turn, this is why things actually aren't "better for nearly everyone" with perfect competition. Pretty much every single product you've ever liked was the product of a very non-perfectly competitive market, from sodas to software to jeans to phones to food. You don't want a generic "hamburger" like every other hamburger in the world, you want one of those great burgers the guy at the burger shop around the corner makes, with extra pickles and garlic aioli, because it tastes fantastic! Or maybe a chicken burger, or hey, maybe even a McDs quarter pounder (I won't judge you).
But the point is, these are all different products, and some people like one, and some like another. Would you be happier in a world with perfect competition in burgers, where every burger was literally identical, interchangable, and indestinguishable from every other?
In fact...would any person be made better off by perfect competition in burgers? Or isn't it actually kind of good that we have choices? That I can start a burger shop if I want to, and offer a range of burgers made with grass-fed angus beef, or whatever? In other words, isn't it good that we don't have perfect competition, so that people can compete? :)
(And, of course, that all goes for software and startups too.)