Installation
1. First, go here: http://tripp.arrozcru.org and where it says in bold, "you just want an FFmpeg executable, get this one", get that one, which is the latest build of ffmpeg.
2. Next go here: http://avanti.arrozcru.com and click download at the top and then download the latest version of Avanti.
Avanti and ffmpeg are stored in the 7z format. If you cannot open these already you will need 7-Zip. I highly recommend 7-Zip anyway. It's free and it does everything WinRAR, WinZip etc can do.
3. So go here: http://www.7-zip.org and get the appropriate build for your system, either 32-bit or 64-bit. If you are unsure, just get 32-bit.
4. Install 7-Zip. When it's done installing, find it in your start menu and open it. Then in the 7-Zip menu bar, find Tools and then Options.... The first tab is called System. Check off any filetypes you want 7-Zip to open, with .7z at the minimum. Here's mine:

5. Open the avanti-039.7z file you downloaded earlier. It's 039 as of this writing, that number may be larger in the future. Unzip that wherever you'd like. C:\Program Files\avanti-039\ will work fine.
6. Next open the ffmpeg zip file. Copy ffmpeg.exe and SDL.dll to the existing folder called "ffmpeg" within the Avanti folder. For instance, C:\Program Files\avanti-039\ffmpeg\.
7. Since Avanti does not have an installer program, you may want to create a desktop shortcut for the exe. You can do this by right clicking the Avanti ("G") icon and hitting copy and then right clicking the desktop and hitting "paste shortcut".
Operation
Now you're ready to start. Go to YouTube, find a video. Copy the URL from your address bar. Then go to http://keepvid.com and paste it where it says URL. Keepvid will typically give you two options, a high and low quality. The audio should be about the same in both files, but I have had problems in the past with the high-quality files not converting completely. Download one and note where you saved it to.
Here's Avanti:

(you may notice that I hadn't put ffmpeg.exe in the right folder before I took the picture
Click the little folder icon all the way to the right of Source 1 and in the dialog that pops up, locate and click on the downloaded .flv file.
Put whatever you want in the box to the right of Destination, I think the easiest thing to do is copy and paste what's in Source 1 and change the file extension.
Where it says Destination Audio Settings, pick mp3 (Lib). You can put what you want for the bitrate. 128 is default, I like 192.
Where it says Destination Video Settings, uncheck Enable.
Finally, click Start process.
There are other ways of doing this, but the basic procedure is to get some software that will decode the FLV and encode the audio stream as an MP3. If you want to edit the file, encode as WAV, make your edits and then use an MP3 encoder (LAME is the standard one) to make the MP3.
I hope somebody finds this useful. Let me know how it works.