How does this inverting opamp circuit with adjustible bandwidth work?
The author is correct in saying that the bandwidth varies with R but the gain doesn't.

This result can be understood easily if we combine the voltage source that is in parallel with R with R itself to get a Thevenin equivalent at the inverting terminal of the opamp.

The Thevenin equivalent will be
$$R_{th} = R_1 ||R$$
$$V_{th} = \frac{V_{in}(R_1||R)}{R_1}$$
And the expression for the gain is
$$A_v = \frac{V_o}{V_i} = -\frac{R_f}{R_1}$$
Which is independant of R.


As the OP has correctly pointed out, the Gain Bandwidth product of an amplifier remains constant no matter the extent of feedback. More on this can be found here and here.

The trick is that the input to the feedback amplifier (inverting amplifier) is Vth and not Vin.

So by increasing R the gain is falling (Denominator increases), since the gain is
$$\frac{V_o}{V_{th}} = -\frac{R_f}{R_1 ||R}$$ 
and consequently, since GBW remains constant, Bandwidth must increase.