Lucky you - I really want one of those Yamaha mixers....they have nice pre amps from what I've heard...
Looks like no USB out, so plug the main outs to an audio interface and then the audio interface to your USB input on your computer.
Tascam, Lexicon, Line 6, M-Audio, are all good choices. You only have L/R out so a 2 channel interface would do. You can also send the mixer inserts to an interface so then a 4 channel interface would work if you want to send 4 mics all at the same time.
Personally, I record one thing at a time, so a 2 channel from the main outs works fine for me.
I use a U42S.