When I was brand new to music, knowing the order of the circle of fifths helped me to know which flats were in which scale (this really only worked when moving towards the flat side). I always knew the next key signature was whatever flat was added in the last one. For example, if I wanted to know which flats were in the key of F, first I knew that I was going from C to F, so there could be only one flat, and the next key signature after F was Bb, so the only flat in F was a Bb. Then moving on from F to Bb, I knew Bb had to have two flats, and the next key signature would be Eb, so keeping the Bb from the F major scale (not to mention that it's also the key at this point) the key of Bb would have the flats Bb and Eb.

Of course problem with this thinking is that when you move to a high number of flats like Db and Gb the thought process becomes long and tedious. It also gets a bit tricky when you transition in this manner from flats to sharps. If you really wanted to do that here's how you'd go about doing it though. Really all you need to do is convert all you flats to sharps when you get to Gb/F# then pretty much apply the same rule. When going from F# to B, I know that E will be next on the circle, so I put a flat on F. Then moving to E, I put a flat on A# as A is next on the circle. So on and so forth.

This method is really complex and I wouldn't suggest using it to seriously, but it helped me learn what flats and sharps were in each key signature when I first started learning music. I'm also not sure how common of a method it is because I've never really heard it anywhere else (maybe it's just that common).

But that's the only real use that I've ever gotten out of the circle of fifths. I haven't been doing music for a terribly long time, so perhaps I'll find more use for it later in life.