As each metronome comes to a stop & reverses direction at the top of it’s swing, it transfers energy to the platform. This energy transfers through the platform to each of the other metronomes.
The energy transferred to the platform from each metronome starts out of sync, but the waveforms of the energy interfere with each other, and at the location of each metronome you end up with negative or positive waveforms which in turn have an effect on the interference pattern. This keeps going until there is a balance – this balance is only achieved when all of the metronomes are in sync. The interference pattern is now a standing wave, which keeps the metronomes locked in phase.