If you want to play the file(s) you downloaded;
1. double-click the file you want to play, or
2. highlight the file you want to play from library>shared (if this is where you store your downloaded files), and click "launch" at the bottom of your window.
If you want to play the incomplete file(s), again, highlight the file, and click "launch". You can do this either in the search window or in the library window.

FYI, LW does not come with a player. It usually uses QT or iTunes to open the files.