Why does a smaller bandgap correlate to increased absorption of visible light?
In your case, absorption takes an electron from a valence band and in to a conduction band. At energies below the narrowest gap between the valence and conduction band, the photon can't give enough energy to an electron to get it up from the valence to the conduction band - there is no absorption there. 

Right at the band gap, you can now just get a small population of electrons from the valence to the conduction band, those right at the top of the valence band. Absorption starts to kick in. 

As the photon energy increases, there is a larger and larger number of valence band states that can now be excited in to the conduction band. 

Depending on details of the shapes of the conduction and valence bands (density of states of initial and final vs energy), and on the exact absorption process (if you need a phonon for momentum matching it is harder), you get various expressions relating the absorption to the band gap ranging from $\sqrt E$ to $E^{2}$ or higher powers.