LAME is a library that allows some programs to encode MP3 files. For more information about LAME in general, click here. LAME is free, but in some countries you may need to pay a license fee in order to legally encode MP3 files.
These files are in the Zip format. Either one should automatically extract if you download it with Safari, or if you double-click it in the Finder.
Once you have unzipped the archive, you will have a file called LameLib or libmp3lame.dylib. To use it with Audacity, you can put it anywhere you want, but the first time you want to export an MP3 file, Audacity will ask you for the location of this file, so you'll want to remember where you put it.