Apparently, a lot of pop music is based on the same four chords, the video does not say which those may be. Any idea what four chords they are talking about?
This chord progression is common enough to have a Wikipedia page:
The 'axis of awesome' video you refer to is talking specifically about those chords in that particular order - I–V–vi–IV - which is a feature of many prominent pop hits. (I'm sure you're aware of this, but for any other readers: just because this is one common chord progression doesn't mean that there aren't dozens of other common chord progressions in pop... as well as some uncommon ones! The idea that all pop songs are based on the same 4 chords is a joke, not a serious assertion.)
Of course those 4 diatonic functions - the Tonic (I), Subdominant (IV), Dominant (V) and Submediant (vi) - are arguably the 4 most important in major tonality, so it's unsurprising that these chords are common, and as b3ko mentions below, that many other popular chord progressions use these 4 chords.
To quote Leonard Cohen: "It goes like this, the fourth, the fifth, the minor fall". Then "the major lift" is another fourth. Those plus the tonic give you your four chords.
If it's good enough for the Lord, it's good enough for pop.
Go to music and harmony theory by Tchaikovsky (I'm name dropping), or any other source. Mel Bay chord melody system for guitar players. The FACT is you can harmonize the entire major scale with just three chords I, IV, and V (or V7 to get the 7-->8, 4-->3 resolution). The vi chord is a viable substitute for the I (I6 and vi-7 are identical). So you see, three chords (and their inversions) is all that is needed! It is inherent in the mathematical structure of the western scales and present in music from the Renaissance era to today. In fact when rock guitarists, or blues, country, etc, make the old joke "All you need to know is 3 chords" I think many folks interpret this as saying modern music is over simplified. In fact the same can be said for Bach. Pop learned from, and borrows from the same old masters as every other style.