Well, this is a kind of a gray zone. The data hosted on sites like Youtube don't belong to them but to the creators. The site itself doesn't download anything, in a way it simulates what your browsers does.
As a user, that's more complicated. In European Union it's legal. Everywhere else? Hard to say.
The site gives you a direct link to the file. Based on the site behavior and your browser settings, your browser will or start downloading or open the video directly.
As a user, that's more complicated. In European Union it's legal. Everywhere else? Hard to say.