OK, I'm trying to type this quickly, since my last essay on this was lost when I got logged off for slow behaviour.
I have Aebersold 65, and 'Four' like all the tunes in here is structured in 'choruses' (five of them of 32 bars in this case).
The 'form' of the chorus starts after the first three notes of the tune, at the double barline. Call the first three notes the 'upbeat' notes or bar 0 if you prefer.
Bar one starts with a double barline with a Colon after it. This is a repeat mark, showing you where to come back to.
Bars 1-4 inclusive are the first line, bars 5-8 the second, and bars 9-12 the third. Bars 13-16 come under a bracket with '1.' in it, this is called a 'first time bar' marking, meaning you only play this fourth line the first time around. Bar 16 ends with another double barline with a colon before it. This means repeat back from here to the 'start repeat' bar (i.e. go back to bar 1 again). Bars 17-20 come under a bracket with '2.' in it, meaning you substitute these 'second time' bars for the 'first time' bars you played last time.
So you play:
Upbeat (after 3 of the count-in)
bars 1-12 (lines 1, 2 & 3)
bars 13-16 (line 4)
bars 1-12 again
bars 17-20
(total 32)
Then repeat this whole 32 bar business four more times (insert the upbeat into the 'break' if you like).
The reason the chords section is written out again is that the chords for the 'solos' section are slightly different than those for the tune. If you want to keep playing the tune over and over (and this would be my preference to start with, then embellish it as you get to know it better and better) it should still fit. The first and second time brackets extend over more of the bars, but they still add up to 32 in total.
To add to the confusion there's also a 'Coda' (another type of jump in the score, last time only, you jump from the first circle-with-a-cross-through-it, like a target crosshair, to the second) which adds a final chord at the end of the 32-bar form.
If you want to go down the route of learning to improvise from chords and scales, you've chosen a fairly meaty book, there are some complicated charts in it. I was recommended to start with Vol 54, 'Maiden Voyage', which has some really simple bluesy and modal tunes with lots less in the way of chord changes, so you can practice a smaller amount of notes to get your ears into gear with the improvising, though Four isn't a bad one. I for one still don't understand 'Giant Steps', but that's a whole new thread...
Good luck,
Nick