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I know that the initial reactants are phosphine and menthanal in an acid ($\ce{H+ A-}$) and that the first step in the reaction is the HOMO of phosphine (the lone pairs) and the LUMO of methanal (the π* orbital).

Then, the electrons that just moved to the oxygen atom react with $\ce{H3O+}$, so the oxygen is protonated. I'm not sure what follows, though.

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The acid serves as a catalyst to protonate the formaldehyde oxygen. This makes the carbonyl carbon more electrophilic, a more reactive target for the weakly nucleophilic lone pair of electrons on phosphorous. Here is a diagram of the basic reaction mechanism.

enter image description here

The proton transfer is thought to be intramolecular according to this link.

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