The interference pattern arises because we are detecting which of two possible states the atoms are in, and this depends on what happens to the atoms in the clock.
If the microwaves are "on resonance", i.e. they are at the exact frequency of the Cesium atoms' natural frequency, then all of the atoms make the transition and end up in one state. This happens at a peak of the interference signal. If the frequency is slightly off, however, there is a phase difference between the two - the microwave oscillations will have gone a little more or less of a cycle than the atoms have, and some of the atoms won't make the transition and will be left in the original atomic state. So the signal decreases. If the phase difference is exactly half of a cycle, none of the atoms make the transition and the signal is a minimum.
If the phase difference is one complete cycle (or two, or three - any integer) then we see another maximum signal. And when the difference is an integer plus a half, we see a minimum.