From my own songwriting perspective:

The song is in E. You've basically got E, A, D with some inversions and bass movement.

But after the sax solo, there is a psychedelic section (D - G - A - C) -- all major chords -- which takes you to another world with unexpected magical-sounding modulations. Where is it going? The whole point of it is to tonally confuse you, build suspense, and to get away from the main key of the song for a bit. That's the whole point of a bridge. We're not in the same world we were in, and where are we going next? Where is the house going to land? Let's find out!

So, it just so happens that the C at the end of that series of major chords (D - G - A - C), if you think of C as a V, resolves to: F! F is also half a step higher than E, which gives you a rising feeling, hearing the chorus played back higher than it originally was. Aha! Exciting.

Then there's the chromatic thing moving down, with the whole band playing it in unison, dropping you back off in the original key of E. This is a common trick. Anders actually covered this same type of idea recently in a GuitarTricks Channel video on YouTube. (Add random chromatics to get from one place to another.)

The point of all of this is: you decide where you are (Key of E), where you're going to go (key of F) and why (to take it up higher as a climax and make it more exciting). Then you fill in the space, connect the dots (by using modulations, chromatics). It's just all tension and release. So that doesn't just answer the "How" but also the "Why."

Someone else might have a different take on this, but in reducing the song down to essentials, that's pretty much what you've got.

-Carl.