The service adds dead simple filesharing functionality to Twitter. Log-in to your Twitter account via TweetCube's site (user names and passwords aren't stored), then upload the file you want to share: images, videos, music, PDFs and more. Next, add a message and hit publish. The result shows up on your Twitter stream with a shortened URL linking to a page hosting the file on TweetCube.
The service is completely free and the only limit is file size (10MB). You can upload as many files as you want, but TweetCube deletes old files after 30 days -- in other words, this isn't a replacement for hosting files on your own Website but instead is designed for 'one-off' broadcasting of files with your Twitter buddies. That's the deal, and at least lessens the risk of using the service in case it's ever shut down :-)