This question is basically all of music theory in a nutshell, right? "What harmonies sound good together?"
It's really all some form of tension and resolution, if you want to simplify it as far as possible. The circle of 5ths is basically a bunch of V-I resolutions stacked on top of each other: In C, you have a C major triad of C-E-G, with a B if you want to include the major 7th, and a G7 triad of G-B-D, with an F if you include the 7th (which for the sake of nailing down why this sounds good you probably should) So, you've got a couple pitches that are constant between the two chords - the G doesn't change since it's the root of your V7 and the 5th of your I, and then if you include your major 7th, the B is a common pitch as well, the 3rd of your V7, and the maj7 of your I. Then, you have two pitches that do move, in opposite directions - the 7 of G, an F, drops a half step to the E in your C chord, while at the same time the 5th of G7, a D, drops a full step to the root of your C. This half and whole step resolution to the tonic is an incredibly powerful sounding resolution and sounds very "final" to your ear.
...final enough, in fact, that you can next treat your tonic as the new V chord, and resolve to another I below it, here, C to F. This is your circle of 5ths.
The "why" though is kind of telling and points to another great way to connect chords - when you go from G to C, and then C to F, and then F to Bb... really what you're doing is G7 to Cmaj7 to C7 to Fmaj7 to F7 to Bmaj7, and there's a chromatic resolution in there that's implied by the harmony even if it isn't actually happening. The B in Cmaj7 drops s semitone down to a Bb, yielding a C7, which then falls another semitone to the A in Fmaj7, while the E in FMaj7 then drops a semitone to Eb, in F7, which falls another half step to D in Bbmaj7. So, what's compelling about this to your ear is that a half step movement in an environment with a lot of other pitches held constant can ALSO sound really consonant to your ear, even when you're technically stepping out of key (I skipped over this, but you're familiar with the harmonized triads from a major scale, right? I, ii, iii, IV, V, vi, viib5, and I? with upper case major and lower case minor?). This kind of resolution can be a lot of fun - in the key of A, for example, try playing a I-IV-V-I in A to really get your head grounded in the harmony, then play A-D-Dm-A. There you have the major 3rd of D becoming a minor 3rd of D minor, then becoming the 5th of A, as a three-note chromatic descent, that links a chord that has no business at all being played in A major, a Dm, into the key (I'm using this as an example because I used this exact move as the resolution for a song of mine once, and thought it was pretty cool).
This is just the tip of the iceberg and to go much deeper I'd want a guitar in hand, but thinking about how chords resolve from one to the next, less because they're a product of a certain scale, and more because of what the resolutions within those chords are, can yield some pretty cool ideas.
If you want some fresh ideas on how to link chords like this, grab a good fake book, with "The Real Book" being the gold standard - sitting down with one and your instrument of choice (here, a guitar) and just sort of strumming through the changes, and unpacking how each chord is resolving to the next can give you some ideas on how to approach non-diatonic harmony in your own writing. And I say this as a shred/rock/blues player and absolutely NOT a jazz cat - harmonic movement is a lot of fun, and what we call jazz today used to just be people doing pop standards and trading solos over them, and you don't even have to use particularly complex chords to take lessons from here (the solo section for one of my songs I'm just wrapping up is straight major chords, but with an implied chromatic descending line stringing them together and linking two distinct tonal sections, which for kicks I ultimately ended up adding a sort of funky strummed/muted rhythm guitar playing a tight chromatic descending octave figure in as a background part under the lead to really bring that out).
