The idea of the circle of fifths is essentially based on this one idea: the musical interval of a fifth is important in music theory.
Let's explore why!
The octave is the most important interval. It is the foundation upon which Western Tonality and Music Theory is built. Measured in hertz the distance from one pitch to a pitch one octave higher is exactly doubled.
The A on the open 5th string of a guitar is 110Hz. The A an octave higher at the 12th fret of the 5th string is 220Hz. These two A's are one octave apart. The interval of a fifth is the musical distance of 7 half steps (or 7 frets on the same string) between any two notes. So, start on the open A string (the 5th string) and regard that as your starting note, your root note. Go up 7 frets and you have the note E. The distance between this A and E is a fifth. These notes are a fifth apart.
Again, the A on the open 5th string of a guitar is 110Hz. The E at the 7th fret of the 5th string is 165Hz. Measured in hertz the frequency of the pitch exactly halfway between any two notes an octave apart is a fifth. Soundwise (acoustically), the fifth splits the octave in half. It is the furthest auditory point away from either A, below or above. This suggests the furthest auditory "place" notes can "move away" from the root note of a scale, and as a consequence, then "return to".
The chord built on the fifth of a scale is labelled the dominant chord because it has this crucial fifth scale degree along with the leading tone, the major seventh scale degree as well. It is crucial, fundamental and pivotal in Western Tonality!
More specifically, it is important to the aspect of music theory known as Functional Harmony. It is important because it is the interval between the root notes of the Tonic (I) and Dominant (V) chords. It is a crucial interval in Western Tonality because it splits the octave in half.