Working Principle of AD8232 Instrumentation Amplifier
GM1 and GM2 are transconductance amplifiers, per the data sheet.  This means that they generate currents that are equal to \$g_m \left ( V_+ - V_- \right)\$, where \$g_m\$ is the transconductance of the amplifier, \$V_+\$ is the non-inverting input, and \$V_-\$ is the inverting input.

The output of GM2 feeds back to the op-amp.  So what is integrated is the sum of:


The non-inverting input to the chip
The negative of the inverting input to the chip
Some constant determined by HPA
The negative of the output of the integrator


That last bit makes the integrator into an amplifier, with a gain fixed by the resistor network that feeds back into GM2.

The reason for all of the folderol with GM1, GM2 and the integrator is almost certainly because the chip designer found it easier to make a nicely matched pair of transconductance amplifiers than to realize the desired functionality any way else.