The major I for the third chord is the really key piece of that chord progression. Shifting up and down chords by step is pretty common in pop music, but shifting into major I the first time then the flat 9 dominant the second time (especially with the natural 7 right before it!) is a powerful harmonic one-two punch.
I don't want to be 'that guy', but the other guy responding to you has it totally wrong. Try to plunk out chords on the piano with the music and see for yourself.
So, to clarify, I'm not claiming there is a III in the original key, I'm referring to the III chord of the relative major key.
For classical music the working key is constantly changing and often two theorists would disagree on the "harmonic meaning" of a passage. One telltale sign of a key change is an x-V-I progression (in the new key) where x is any secondary chord that leads naturally into the V. Bonus points if the x or the V don't work in the old key; if you're in C major and you hear a D7, you know something's up.