Translation:You like to listen to your son play the flute.
It's the same as English. To listen and to hear have different meanings. It may come down to the same thing but it is different.
To hear represents automatic or passive perception of sound, while listen generally represents intentional or purposeful use of the sense of hearing. A similar distinction exists between see and watch.
The finite verb is piace. Everthing else is an infinitive. In English we say "he plays the flute" but "I like to hear him play the flute." You could just as well ask "why isn't it 'plays'?" The answer is the same, although non-linguists don't usually call it an infinitive without the "to" in front of it.
Am I the only one who hears the voice say "suo figlio" instead of "tuo figlio"?