Learn Italian: Sento una voce, ma non c'è nessuno. - I hear a voice, but nobody is there.

Yes, and Italian loves double negatives! When negative pronouns like nessuno (nobody) or niente (nothing) come after the verb, you must put non before the verb. If you started the sentence with the pronoun, you would drop the non (like Nessuno c'è), but non c'è nessuno is the most natural way to say this.

The English translation says 'nobody is there'. Where is the word for 'there' in the Italian sentence?

It is built into c'è, which means 'there is'. The phrase non c'è nessuno literally translates to 'there is not nobody' (meaning 'there is nobody'). Italian uses c'è for this instead of adding a location word like lì or là.

I learned that sentire means 'to feel'. Why does it mean 'to hear' here?

The verb sentire is used for several senses. It can mean to feel, hear, smell, or even taste. You just have to use the context to know which one it is. Since the object is una voce (a voice), it clearly means 'I hear' in this sentence.

Could I say ascolto una voce instead of sento una voce?