The websites have to enable AirPlay on their end, by default it is turned off. As a user this isn't great but I suppose the publisher should have some control over how their content is displayed.
If you want a website to support AirPlay you're going to have to ask for it. Don't accept a "it's too complicated" answer either, turning AirPlay on is trivially easy.
Edit: There is also a workaround; check out
this bookmarklet that will enable AirPlay on any HTML5 video tag.