Of course there are infinite ways one might explore a scale or harmony compositionally, but one aspect of the pitch collection that made it interesting to Scriabin is that it can be used to resolve in a more-or-less traditional manner to a number of different distantly related harmonic areas.
First off, a reminder about the harmonic possibilities of a standard dominant seventh chord (and let's leave out the fifth of the seventh chord for reasons that will become evident later). Say we have C, E and Bb. This could be a C7 chord, which implies a tonal resolution to F. The most important melodic driver of that resolution is the tritone between E and Bb, which wants to collapse (or expand if written as an augmented fourth) into F and A. Of course, any tritone can be enharmonically reimagined. In this case, we could also see it as E and A#. This tritone wants to expand (or collapse if written as a diminished fifth) into B and D#. Thus E and Bb/A# could either be used for a melodic resolution to F or to B which is a tritone away and as far apart as any two major keys can be.
But the C7 harmony we were discussing also has a harmonic reason to resolve to F--the movement of a fifth/fourth from C to F. We could use it to resolve to B, but it wouldn't be as convincing in tonal terms. The notes E and A# would need a root below them that could turn the harmony into a dominant seventh (with a missing fifth). That note would be F#, giving us F#7--F#, A# and E. If you look at the Prometheus scale, you'll see that it also has F#, so a composer is able to construct either C7 or F#7 just as easily. Furthermore, the entire collection played as a harmony--Scriabin's so-called "mystic chord"--has equal potential to resolve in either direction.
Notice however, that the two roots C and F# are also a tritone apart from each other, and thus could be used to melodically resolve to G or (with enharmonic respelling of F# as Gb) to Db. The first of these possibilities is further strengthened by the presence of D and A in the chord, which could be heard as the root and fifth of a D7 chord--D, F#, A and C. When used as a harmony, the chord has equally clear tonal possibilities to resolve in F, B or G, making it into a sort of asymmetric three-tonic system.
This kind of ever-shifting quality is common in a lot of Scriabin's music, which often sounds surprisingly tonal despite some very dense harmonies and unorthodox connections. Scriabin also used the collection as source material for other harmonic sets and melodic figurations. Its prime form is (013579), and a lot of interesting harmonies can be picked out of it. It's almost, but not quite, symmetric (essentially a whole tone scale with one note altered), and that means that it has many transpositions and inversions that will retain several common tones. This can be useful for creating transitions and connections between different harmonic instantiations. However, because it isn't truly symmetric, no transposition creates the same harmony exactly, there will always be at least one different pitch, which allows for greater variety. A deeply interesting pitch collection.
