There are a a few ways to go about this, if I understood your post correctly.
As already suggested by atown44 you could simply record the sound output of your computer. This can be done using many different programs available on the Internet. There are plenty of free sound recorders out there that you can use. Personally when I wish to record sound output I use
Audacity. This involves setting your "input" on volume settings to "Stereo Mix" (I've also seen it called "WhatUHear", and most likely it can be named other things depending on the maker of your sound card) and then simply pressing the record button and exporting the recording to an MP3 file.
If the flash file in question is an animation (such as those found on
http://www.weebls-stuff.com/) then the actual output is more than likely made up of a few different sounds embedded within the swf that play at a specified time. These can be recorded using the method I have stated above, or if you save the swf file on your computer the individual sounds (e.g. background music in the Weebl & Bob toons) can be ripped using software, such as
Flash Decompiler. I don't know of any free SWF decompilers though, so if anyone else has any links feel free to post them.
Instead of ripping each individual sound in cases like the above, you could also run the swf file through a converter such as
SUPER and selecting .mp3 as the output format, which would give you an mp3 recording of the flash file.
For sites that have a player that loads different mp3s when you select them from a list or something, there is another method, however it is much more complicated, and involves trying to find out where the flash file loads the mp3 file from. This involves extracting the action script from the flash file, and finding the part that actually loads the mp3, and then using that to work out the complete URL of the .mp3 on the server. If you do go down this route, one freeware action script ripper I use is
Flare.
Personally I'd go with the first method or the second, depending on what it is you wish to extract the mp3 from.
Hope this helps