How does PTO work in the US?

European here with a stupid question for the Americans:

I've seen a lot of reels on Instagram on people not having their PTO requests accepted for weddings and other important occasions. And a dad saying that he was ill but had no PTO left so he had to go to work.

The labor laws of my country allow for workers to be off sick for 2 whole weeks before having to document anything to their employer (and only if asked).

Not posting this to belittle anyone - just wanting to understand the workings of a different system and realizing that I maybe am going of with some generalization here...
(And sorry for any language mistakes!).