Could someone help me understand this simple circuit?
Maybe look at it like this: -


I've removed components that get shorted or open circuited by the PTT switch contacts

In transmit mode, the speaker becomes the mic and the op-amp has a large gain determined by R3 (100k) and the small impedance of the speaker. The Rocking Armature's impedance is 150 ohm so for a rough estimate, the gain is 100,000 ÷ 150 = 666. This means that a 10mVp-p signal on your "mike" gets amplified to 6.67Vp-p.

In transmit mode R4 is superfluous but I left it in because it wasn't exactly open circuited or short-circuited by the PTT switch. C1 in transmit mode is just stabilizing the non-inverting input of the amp.