By "verb phrase," I think you mean clause? (A phrase, or string of words, that includes a verb is a clause.)

You can use synchronize as a verb:
The swimmers synchronized themselves with the music. The swimmers are actually performing the verb.

But you wouldn't say, for example, "the music synchronized itself with the swimmers" because the music can't synchronize itself, right?

Those are examples of active voice - the subject of the sentence must actually perform the action of the verb.

The subject in this question is the "movements." Can movements synchronize themselves? No - the swimmers could synchronize themselves, but the swimmers are not the subject. Because "movements" cannot actually perform the action of synchronizing themselves, we have to use passive voice here: the movements are synchronized (something else is causing the synchronization - in this case, the swimmers).

Passive construction consists of some form of the verb "to be" (in this case "are") followed by the past participle of the action under discussion (in this case "synchronized"). When you use this construction, the subject is not performing the stated action; something else is performing the stated action ON the subject. (In this case: the swimmers are synchronizing their movements.)