Learn Russian: Он попросил прощения у сестры.

Verbs of asking, wanting, and waiting (like попросить, хотеть, ждать) often take the Genitive case rather than the Accusative when the object is an abstract concept, like forgiveness. If he were asking for a specific, tangible object, you would use the Accusative (for example, он попросил книгу to mean "he asked for a book").

To say he asked "his sister", why do we use у сестры instead of the Dative case (сестре)?

In English, you apologize to someone, but in Russian, when you use попросить (to request or ask for), you ask for something from someone. This requires the preposition у followed by the Genitive case. So, the literal structure is "He asked for forgiveness from his sister."

The English translation says "his sister's," but there is no word for "his" in the Russian sentence. Why?

In Russian, possessive pronouns are frequently dropped when the relationship or ownership is obvious from the context. Since he is asking for forgiveness from a sister, it is naturally assumed to be his own. Adding своей (to make it у своей сестры) is grammatically fine but completely unnecessary and sounds slightly less natural.

Could I just say Он извинился перед сестрой? What is the difference?