I'll keep this as simple as possible. The gain of the 1st stage is determined by the impedance of everything between pins 2 & 3, which we'll call Zf, and the impedance of everything between pin 3 & ground, which we'll call Zg.
The gain, G is given by:
G = 1+ Zf / Zg
Notice this if Zg gets smaller, the gain gets larger. The Boost switch puts R15 & C15 in parallel with R12, R13, C12 & C13. Engaging the Boost switch makes Zg roughly 3x smaller, which will make the gain approx 3x larger.
Because of the capacitors and diodes, the gain varies with frequency & amplitude, but the general principle is still valid. Note that this is a gain boost and not a volume boost. The apparent volume will go up a little when Boost is engaged, but the main effect of Boost is more distortion and sustain.