Basically, this application can make YouTube videos (among others from different online sources) look fancier on your browser window. Thus, it's not really a YouTube competitor as YouTube is one of the potential sources for content.
Using this tool on a site actually sounds more complicated than it is. Publishers and developers can add add the WebPlayer to a particular site with one line of code. Once that is done and the page loads, the player scans the page's contents and automatically inserts an obvious play button next to supported and playable items. (That's usually links with movie titles, etc.) After clicking on the play button, the WebPlayer slides out at the bottom of the browser window for playback.
For a better idea, check out the sample below, or view some of the samples that Yahoo has posted.
The idea here is that it should be much easier for bloggers, publishers, or whoever that wants to publish digital media on their websites on a regular basis as all they really need a line of code and a link rather than copying over a giant embed code from YouTube.
WebPlayer is still in beta mode only, so there are surely going to be some quirks and incompatibility issues until the full version is ready.