I agree with Freancesco that for a detailed answer you should read Farb/Margalit, but for an executive summary:

The mapping class group is generated by Dehn twists, the symplectic group by transvections (elementary matrices). Dehn twists get mapped to transvections by the Torelli map (your $\phi.$)

I agree with Francesco that for a detailed answer you should read Farb/Margalit, but for an executive summary:

The mapping class group is generated by Dehn twists, the symplectic group by transvections (elementary matrices). Dehn twists get mapped to transvections by the Torelli map (your $\phi.$)