How does this OP amp circuit with a transistor work?
Here's a very similar circuit from the datasheet of TI's LMP8646:



Note that in the feedback loop, the transistor inverts the feedback, so that even though feedback connects to the non-inverting input, it's still negative feedback. 

In order to equalize the voltage of the two inputs, the control loop will need to pull enough current through RIN to create a drop across RIN equal to the drop across the shunt. This current will be equal to \$I_{\mathrm{sense}}\frac{R_\mathrm{shunt}}{R_\mathrm{IN}}\$, so usually a pretty small value. 

In the TI circuit, essentially all of this current will flow through the transistor to RO, and the voltage at VRG will be $$R_{\mathrm{RG}}I_{\mathrm{sense}}\frac{R_\mathrm{shunt}}{R_\mathrm{IN}}$$

The buffer will just reproduce this same voltage at the output with lower output impedance.

Note that in your circuit there will be a slight difference due to the BJT adding some base current to the output.