I'm trying to figure out what a Java applet's class file is doing under the hood. Opening it up with Notepad or Textpad just shows a bunch of gobbledy-gook.
Is there any way to wrangle it back into a somewhat-readable format so I can try to figure out what it's doing?
what you are looking for is a java de-compiler. I recommend JAD http://www.kpdus.com/jad.html It's free for non commercial use and gets the job done.
Note: this isn't going to make the code exactly the same as what was written. i.e. you're going to lose comments and possibly variable names, so it's going to be a little bit harder than just reading normal source code. If the developer is really secretive they will have obfuscated their code as well, making it even harder to read.
JAD is an excellent option if you want readable Java code as a result. If you really want to dig into the internals of the .class file format though, you're going to want javap. It's bundled with the JDK and allows you to "decompile" the hexadecimal bytecode into readable ASCII. The language it produces is still bytecode (not anything like Java), but it's fairly readable and extremely instructive.
Also, if you really want to, you can open up any .class file in a hex editor and read the bytecode directly. The result is identical to using javap.