It means that the value can not be changed later. So in the example above ANGLE_LIMIT will always equal 135 and statements that attempt to change it are rejected e.g.

ANGLE_LIMIT = 90; // Syntax error

These are normally refered to as constants and by convention constanta names are always uppercase.

And as a bonus, final field variables are faster! Especially when paired up w/ static!  

GoToLoop, you should try and avoid to propagate myths about Java performance...
There are old articles telling that sprinkling final everywhere helps the compiler to optimize the code, but if they were ever true, it was for old versions of Java. Now, the compiler is much smarter than us (for these topics), so it doesn't need our help with such hints.
I also saw some articles debunking these myths...

In short, use final where it makes sense (creating constants) or where it is needed (access from an inner anonymous class, for example), not for supposed performance.

Is there any real advantage to making something final? Would it just be to make it clear to other programmers that it won't change?

Security: if you make a class, you don't want users of your library to change your constants!
Documentation: a final class indeed marks it explicitly as constant.
Convenience: if a variable is static final, Eclipse can shows it value by hovering over it.

Etc.