Learn Italian: Il nonno è vecchio e non ci sente. - Grandpa is old and he can't hear.

It changes the verb sentire (to hear) into the pronominal verb sentirci (to have the physical ability to hear). Instead of saying he isn't hearing a specific sound right now, non ci sente means his actual sense of hearing is failing.

How is sentirci similar to volerci and metterci?

They are all pronominal verbs! Just like adding ci to volere or mettere creates a brand-new meaning about time duration, adding ci to sentire creates a new meaning about physical ability. The ci does not mean "here" or "us" in these verbs; it is simply a built-in particle that changes the dictionary definition.

Could I say non può sentire instead to mean "he can't hear"?

You could, but non ci sente is the most natural way Italians express that someone is deaf or hard of hearing. Non può sentire sounds more like he is temporarily unable to hear something in this exact moment, perhaps because the room is too noisy.

Why is there an article in Il nonno? I thought family members don't use articles.