The Anglo-Saxons had strict codes of conduct related to their fellow "soldiers" and battle-mates. Being "heroic" meant living by certain codes of conduct.
Honor was one of the most important codes that Anglo-Saxon warriors lived by. They chose to live and fight with great honor and sacrifice for their fellow "brothers-in-arms." They fought until the death for their brethren. They believed that in death in battle, one made the ultimate sacrifice for their beliefs. The bond that these warriors had with their fellow "soldiers" was often stronger than ones they had with their families, as well.
Being heroic also meant loyalty to one's fellow countrymen and "soldiers." To not be loyal was to punishable by death or banishment. eNotes states:
Loyalty is one of the greatest virtues in the world depicted in Beowulf. It is the glue holding Anglo-Saxon Society together, but it brought with it the darker duties of vengeance and feud. ("Beowulf Themes")
Loyalty also included, as the quote references, seeking vengeance for wrongs done against them...murders of loved ones, etc.
Another important code of conduct involved bravery. These warriors didn't back down from dangerous battles and "missions." They were expected to never run from a fight and to fight until the death if need be.
Beowulf seeks personal glory, as well, for his acts of valor and bravery, etc., and he has no problem bragging about his own abilities!
All of these encompass what the Anglo-Saxons would have considered a true "hero."