The Circle of Fifths can be seen as the gateway to understanding music theory.
The circle is the concept that all western music is based on. it illustrates the relationship between scales in a way that is, hopefully, easiest to understand.
Think of the Chord Progression I - IV - V - I
It's an example of subdominant (IV) and dominant (V) chords leading back to the tonic (I). If we have a tonic note we can use the circle of fifths to give us the subdominant and dominant chord of that key.
Just locate the tonic in the circle, let's use C as an example, then locate the keys immediately adjacent on each side. So, if we're using C as the tonic the subdominant would be F, immediately counter-clockwise, and the dominant would be G, immediately clockwise.
Patterns like that are what make the circle of fifths interesting. Not only does it make it easy to figure out chord progressions in any key it also creates a blank canvas for you to come up with your own chord progressions by experimenting with different patterns. Many musical styles are determined by the shape in which the chord progressions make on the circle of fifths.
For instance, if you draw an equilateral triangle with each point pointing to a chord you'll get a progression where the chords are all a major third apart.