How does this seemingly atonal piece by John Williams still sound so logical?

At 1:48 it starts to be a little bit more tonal albeit still chromatic.

Sorry if this was a rhetorical question; I'm gonna answer it anyway since it's something I would have wanted to know as a "young" composer!

Basically, because John Williams is awesome! I think a lot of it also comes down to recurring intervals and motives (that little lower-neighbour, almost-Dies-Irae thing is everywhere, most measures seem to start with a descending semitone, and the vast majority of intervals are either semitones or minor 3rds, especially early on). To my ear, it also seems like there are also lots of octatonic sets (maybe he switches between them by common tone? Not really sure.) A lot of it comes down to repeating and transposing short, simple, almost-tonal fragments in unexpected-but-consistent ways.

I think having a fairly simple rhythmic profile and discrete phrases also helps--there are still lots of patterns for the ear to latch onto, even if they're not as obvious as Mozart. I'm personally reminded a lot of Bartok by the first half of this piece, but maybe that's just me.

This is a really interesting and unusual score. I discovered that that first cue is largely made up of differing forms of octatonic sets which modulate to the two differing transpositional sets. He also mixes different scale forms (see Bartok violin sonata 1 and 2). It’s not atonal in this sense but rather 12 tone which is not quite the same thing and falls under the term Polymodal Chromaticism. This means combining different scale forms and the modes that can be drawn from them. 
 

JW’s chromatic technique is not unique and I think lends itself more to Bartok’s use of pitch organisation then any strict serial approaches. The Dies Irae pattern recurs in a number of JW’s scores of course as does other familiar chromatic  patterns. Some of the melodic directions or motifs are repeated and developed (changed) as you’d expect…