Why does the voltage gain of a transistor-based amplifier vary with input frequency?
Response is low at low frequencies because \$C_1\$ blocks low frequency signals from reaching the input of the amplfier.
Response is low at high frequencies because of parasitic capacitances within the MOSFET. A typical small-signal model of a MOSFET looks like this:




(Image source: QSL.net)

Typically, the most important parasitic for limiting the high-frequency response is \$C_{gd}\$. The gain in the circuit multiplies the effective value of this parasitic, by what's known as the Miller effect. 

The main effect of \$C_{gd}\$ is to provide a negative feedback from the output node to the gate node.

Cds can also be important. In an ac equivalent circuit, it is essentially in parallel with the output capacitor (\$R_D\$ in your diagram), reducing the overall output impedance, and thus the gain, at high frequencies.