Beowulf wants glory because it is through performing great deeds of strength and courage that he will be remembered. Living on in stories (like Beowulf!) is the only form of immortality available to these pagans, and forms their highest goal. In fact, the real problem Hrothgar faces is not so much the murderous Grendel, but the loss of face he suffers by being powerless to stop the monster. Beowulf‘s reputation, which Hrothgar knows first hand, makes it possible for him to ask that he alone face Grendel; it is because of his former glory that he can pass this greater test. Unferth’s challenge, in which he asserts that Brecca was able to defeat him in a swimming contest, is important because it calls Beowulf’s stature into question, but it is significant to note that Beowulf’s response is one that–although he could have beaten Brecca—he chose (out of friendship and loyalty) not to leave him. Beowulf then compares his glory to Unferth’s, saying, in effect, that he has never heard any such stories about Unferth. Whether or not these fantastic stories about Beowulf are true, the fact that such stories exist make it possible for Beowulf to achieve even greater glory.

Although Christian themes exist alongside pagan themes in Beowulf, a warrior might hope to obtain immortality by performing brave deeds and thus, live on in stories and human memory. Living on in history and human stories, not necessarily heaven, was the glory of immortality. 

And while Beowulf performed heroic deeds for glory, he also did so out of a sense of duty. On one hand, this was out of a sense of duty to Hrothgar. On the other hand, one could argue that Beowulf performed heroic deeds as if it were his fate (wyrd) or a spiritual calling. 

That being said, in the context of the Beowulf manuscript, being a good king meant being kind to his people and being a good warrior meant loyalty to his lord. So, Beowulf was a brave warrior because he did want to be remembered as a courageous warrior and later, a generous king. But he also acted out of a sense of duty, loyalty, and purpose (this could be a moral code, a spiritual inkling, or a personal drive). 

Why did Beowulf become a hero?

In the Scandinavian society into which Beowulf is born, a son is expected to follow in his father's footsteps.  In this case, Beowulf's father, Ecgtheow, is a great Geat chieftain.  As Beowulf describes him to Hrothgar's coast-guard:

My father was well known to folk,/a noble chieftan by the name of Ecgtheow;/ he tarried many winters, ere he turned on his way,/old in years; any wise man/ readily remembers him around the Earth. (ll. 262-266)

If Beowulf behaves as his society expects, therefore, he is born to be a hero, a leader, a son who will reflect all the virtues of his father and family.  In addition, Beowulf is the nephew of Hygelac, one of the Geats most powerful kings, and as part of Hygelac's immediate family, Beowulf would be expected to embody the heroic virtues of both his father Ecgtheow, who married Hygelac's sister, but also those leadership qualities directly attributable to Hygelac himself.  In Scandinavian and, later, Anglo-Saxon societies, the bond between uncle and nephew is as strong as that between father and son.

In Scandinavian and Anglo-Saxon societies, the line of succession among kings and other leaders is from father to son.  If that line is broken by a son who fails to exhibit the necessary physical strength and appropriate heroic virtues, the society is at risk because there is then a gap in leadership that creates chaos.  

If we were to perform some psychoanalysis on Beowulf, then, we would conclude that Beowulf--as the son and nephew of great leaders and warriors--is born to be a hero.  Because family is all-important in these societies, in a sense, Beowulf has no choice but to become a hero.  To become anything less, in this society, would be disastrous for both Beowulf as an individual and the society as a whole.