How is it described in Beowulf how he (Beowulf) becomes the king of the Geats?
Chapters twenty-nine through thirty-one describe of Beowulf describe how Beowulf came to be the king of the Geats. Upon returning from the Danelands, Beowulf goes straight to his king, Hygelac, and presents his rewards from Hrothgar to him. Hygelac recognizes the greatness in Beowulf (given his renown, lack of cruelty, and possession of great strength). Knowing Beowulf's promise as a leader, Hygelac presented Beowulf with a noble sword, "seven thousand hides of land, with a large house and a seat of authority." This presentation proved Boewulf's worthiness to the throne and, essentially, served as Hygelac's naming of Beowulf as the next king.
Upon Hygelac's death (which came during a raid), the Danelands "came into Beowulf's hands."