I have it, it's a really good device but it's purpose is only for streaming content, it itself does not have memory.
Certain apps are supported, those that are (like Netflix) have a button that send the stream to the Chromecast. It also supports the Chrome browser so any videos you can load in there can be sent to the Chromecast. You then control the playback from the device that sent the stream.
As far as how it works, it doesn't send the stream through your phone to the Chromecast, but rather when you send it over it basically tells Chromecast where the video is and then the Chromecast downloads and streams it.