Learn Russian: Мой брат не слушает этих певцов. - My brother doesn't listen to these singers.

It is actually in the Accusative case! The verb слушать (to listen) takes a direct object in the Accusative. However, because singers are living people (animate), their Accusative plural endings copy the Genitive plural exactly. This is why эти певцы becomes этих певцов.

How do we get певцов from the dictionary form певец? Where did the е go?

The noun певец has what is called a fleeting vowel. When we add grammatical endings to it, the last е in the stem drops out. The stem becomes певц-, and then we add the animate Accusative plural ending -ов to make певцов.

In English we say listen TO these singers. Why is there no word for to here?

In Russian, the verb слушать connects directly to its object without any prepositions. You just put the person or thing you are listening to right after the verb in the Accusative case. Think of it more like hearing a singer.

I remember that negative words like не sometimes force the object into the Genitive case. Is that happening here?

Great question, but no! While the Genitive of Negation is a real rule you will learn more about later, you do not need it here. Because певец is animate, its Accusative plural always looks exactly like the Genitive plural, even in a positive sentence like Мой брат слушает этих певцов.