I think it's because of the conservation of angular momentum moving the cans, but I'm still confused as to why the angles between the metronomes must be approximately equal after a short amount of time. I understand that it has to do with coupling effects and that they transfer energy in between the metronomes, but I'm having difficulty finding what coupling actually means.
You probably also have to consider resonance too, which for a forced linear oscillator means that the amplitude of the oscillator can increase a lot when the forcing frequency is near the resonance frequency, which for the depicted setup means a particular metronome will force other metronomes with a higher amplitude when it moves in synchronization with the common (forcing) board. For the metronomes to also change frequency to synchronize across each other you need them to be (at least) a bit non-linear so that the frequency of the metronome is coupled with its amplitude.
In general, you can expect find mode-locking dynamics in almost any set of coupled non-linear systems, but I must admit that this is a very neat and classroom-friendly example of such.
Probably wouldn't work with my wife's electronic metronome though
