Well, first of the cyanide addition isn't an aldol process, it's a Strecker type addition, but regardless it's okay up to that point. The key is the once the cyanide adds into the aldehyde you have what amounts to a benzyl alcohol derivative, which is again oxidized by MnO2. The product is an acyl cyanide, which is quite reactive (see Mander's Reagent). It turns out that CN- is a better leaving group than MeO-, so methanol exchanges in a process similar to trans-esterification.
The mechanism for oxidation by MnO2 in itself is pretty interesting. Anyone care to take a stab at it?
Mander's reagent reference: Tetrahedron Lett. 1983, 24, 5425-5428.