Depending on the status and configuration of your comp (HD usage, CPU usage, RAM available, blah blah...) the LW player might take a while before it actually plays a song (or anything else for that matter) after you click Launch on a file. Sometimes it might even skip during playback (I'm hoping that this won't be that much of an issue in the latest version of LW as the player has been improved).
Anyway, try clicking a song file and waiting for awhile.
If nothing happens, try doubleclicking the file (not Launching it with the Launch button). Again, wait a while.

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iMac G4 OSX 10.3.9
RAM 256MB
LW 4.10.5 Basic
ADSL anything from 3 to 8Mbps/around 1024kbps

"Raise your can of Beer on high
And seal your fate forever
Our best years have passed us by
The Golden Age Of Leather"
-Blue Öyster Cult-