I found an excellent video series on music theory made by Michael New in youtube. That series was the first one that explained me how the theory applies and what does it mean.
The theory forms a taxonomical system to differentiate between musical pieces. The differentiation can be used to produce new music. It's the same reason why I had to come up with the game mechanics taxonomy.
Brilliantly good ideas don't tend to make a finished song or game, they need mediocrity to accompany. If you have a taxonomy in place, the taxonomy can be used to fill in the blanks.
Why does music theory work?
Music theory relies on the idea that human hearing groups same notes within different octaves together. C3 is similar to C4 and C5. Therefore the note can be transposed between octaves with little changes to how it sounds.
I guess it happens because the harmonic content of instruments do not group to just one frequency. They sound with several frequencies at once. You tend to get 220Hz with 440Hz and 880Hz, and possibly some more. The division into octaves represent that fact. The A3=220Hz, A4=440Hz and so on.
The existence of octaves turn the musical scale into cyclic. It limits the amount of unique chords you can form. Therefore amount of chord progressions are limited.
The chord progression doesn't only catch to songs with chords. It applies to melodies as well. Although you can't switch one note in a melody with an octave without it being noticed, the combination of notes played within a melody tells how it blends with a chord. Therefore chord progression exists for every song and not just the ones with chords in them.
The chords themselves - the fifths, major and minor thirds are related to concept called consonance. Any audio you hear is a sum of frequencies. If the frequencies are close enough, they interact differently than if they were apart.