How to analysis this Operational-Amplifier circuit?
The overall circuit is a 2-amplifier Instrumentation amplifier, as Neil describes.

The role of \$R_1,R_2,C_1\$ is to maintain the stability of amplifier IC1 across all frequencies where IC1 has gain (well above the audio band).

When IC1 is driving a high-impedance load it operates as a unity-gain non-inverting buffer amplifier with feedback from the output taken through \$R_2\$.

When IC1 is driving a low-impedance or heavy capacitive load IC1 sees only the isolation resistor \$R_1\$ and feedback is taken through \$C_1\$.