In figure 2 I show the ideal response curve of a typical transmitter cavity. Note that it passes almost all of the signal on the transmit frequency of 145.37. The two receiver cavities are exactly the same except that they have their pass band tuned to 144.77 the receive frequency. Their notch or Band Reject frequency is on the transmitter frequency of 145.37. Thus the transmitter cavities keep the wide band RF noise from the transmitter out of the receiver, and the receiver cavities keep the RF power from the transmitter out of the receiver.
Some transmitters radiate less white noise then others.
A few very early solid state transmitters produced so much noise that they could not be used on repeaters. Electrically they are very simple. One tuned circuit sets the pass band, the other sets the frequency of the notch. Two cavities are used in series simply to provide more isolation between the transmitter and receiver. One cavity isn't enough to do the job.