Learn Russian: Анна сейчас отправляет письмо брату. - Anna is sending a letter to her brother now.

Брату is in the Dative case, which is used for the indirect object (the person receiving something). Since Anna is sending the letter to her brother, the masculine noun брат takes the -у ending.

Why didn't письмо change its ending if it is the thing being sent?

The thing being sent (the direct object) takes the Accusative case. For neuter nouns like письмо, the Accusative case looks exactly the same as the basic dictionary form.

The English translation says 'to her brother'. Why is there no word for 'her' in the Russian sentence?

In Russian, possessive pronouns like 'her' or 'his' are often left out when it is obvious from the context. Since Anna is doing the sending, it is naturally assumed she is writing to her own brother.

Does отправляет mean 'sends' or 'is sending'?

It means both! Russian does not have a separate 'is doing' tense like English. The word сейчас (now) makes it clear that the action is happening right at this moment, so 'is sending' is the most natural English translation here.