Control transmitter operation
For a control transmitter, I know that it converts the shaft position
to an electrical signal. How exactly does it do that?

Do you understand 3 phase motors and generators? If you don't then any answer is going to be too lengthy to fit the format for this site. However, Moog do a very good pamphlet on them so I would urge you to read it and learn. There is a lot to take in but, in a nutshell: -


If this signal were to amplified, how is the control transmitter
connected to an op-amp?

These sort of 3 phase synchro-resolver transmitters are not connected to op-amps; they are usually connected to an identical device via S1, S2 and S3 thus movement of one device causes movement to the connected device and vice versa (this is why you need to understand 3 ph theory). Sometimes the three outputs (S1-3) are connected to ADCs and the angle is derived using an MCU.
I have never seen them used with op-amps because the output voltages don't need amplification.