In Slonimsky's book, page 27 (page 39 on your pdf) "Ditone Progression, Equal Division of One Octave into Three Parts" you see that Slonimsky uses the notes C E G# and C again. They all are a major third apart and they divide the octave into three equal parts.
Now let's look at the sheet music of Giant steps:
The piece has 3 different tonalities. It starts off in B major, it then goes to Eb major and finally to G major and all over again.
If you notice, all of these tonalities are a major third apart*, like the scale Slonimsky mentions. So basically, Coltrane took what Slonimsky created as a scale and used it as different tonalities.
I've heard this rumor as well, read about it as well, but I'm not sure if Coltrane ever actually claimed this, or if people just deduced it. The rumor does make sense though.
*okay, ascending a major third from B you get D#, but its enharmonic equivalent is Eb. It's easier to play in Eb rather than in D#. Also another possible reason might be that the major third of D# is Fx, and the major third of Fx is Ax and it just gets confusing; it's easier to substitute D# with Eb