Learn Hardcore Swedish: Erik ringer sin bror.

Because sin/sitt/sina is the reflexive possessive that refers back to the subject of the same clause. Erik ringer sin bror means Erik calls his own brother. If you say Erik ringer hans bror, it means Erik calls another man’s brother (a previously mentioned male, not Erik himself).

What exactly does ringer mean here?

It means “calls (on the phone).” The verb ringa covers phone calls. It can also mean “to ring” (as a bell/phone makes a sound), e.g., Det ringer = “The phone is ringing.”

What tense is ringer? Can it mean “is calling” as well as “calls”?

Yes. Ringer is present tense, and Swedish present usually covers both English simple present and present progressive:

Erik ringer sin bror = “Erik calls/is calling his brother.”

Do I need a preposition like “to”? Should it be ringa någon or ringa till någon?