In an inverting amplifier configuration why does do the gain resistors affect the circuit's frequency response if their ratio stays consistent?
The resistors are acting as the op amp's load in the absence of any other load.

When there is a larger load (smaller value resistors) the op amp's output stage transistors must be driven harder to supply the extra current and still keep the output voltage at the voltage that it should be. This means that extra current is required to flow back and forth between the op amp's differential input stage and its second stage (output stage driver). The only way to get this extra current is if feedback causes a bigger difference voltage between the inputs. 

Now, the difference voltage between the inputs is caused by the output lagging behind the input signal and there must be extra output phase lag to create a bigger input difference signal which causes a larger current out of the input stage which drives the output transistors harder to supply the load and maintain output voltage accuracy.

Like any single pole (low pass filter) phase lag is unavoidably linked to output amplitude and therefore the extra output phase lag causes an inevitable reduction in output amplitude and a resulting reduction in bandwidth.