I know how to download flash videos. Is there a similar way to download pure audio easily?
Audacity has a feature that will let you record all the audio output from your PC's speakers (this doesn't extract directly from the Adobe Flash formatted file, which could be problematic anyway depending on how the audio is included in the Flash file, but it will let you record the sound -- hopefully that's what you're really trying to achieve):
Audacity (free and open source)
IMPORTANT NOTE: There is a small glitch in the interface that when you select your audio input device that Audacity sometimes selects it for the whole system, so you'll want to take care to note what the original setting was (before changing it) so that you can change it back after you've finished recording.
Additionally, if you want to save your recording in MP3 format (which I suspect you probably will want to), this web page will also be of interest to you as it explains how to install the LAME plug-in (which handles MP3 output):
MP3 support in Audacity