As your question suggests, the motives for Beowulf's decision to risk his life and the lives of his warriors are complex and cannot be attributed to a single cause. In Scandinavian and Anglo-Saxon warrior cultures, for example, leaders and their followers fight for personal glory, for loyalty to family or allies, for treasure, for dominance of territory. Of these, the first two are the most compelling. Because tribal warfare is so common in these societies, being skilled at all things related to war is the only guarantee of survival, and loyalty among tribes and between individuals is the only guarantee of achieving survival and sustained peace. When Beowulf decides to leave his tribe to fight Grendel for the benefit of Hrothgar and his Danes, Beowulf's most powerful motives are the quest for personal glory and loyalty to someone who is part of Beowulf's extended family.
When Beowulf and his men arrive in Hrothgar's territory, and Beowulf is challenged by the Danish coast guard, Beowulf states his purpose:
I through my mind's range/may advise Hrothgar how he might,/the wise and good man, overcome the fiend [Grendel],/if a reversal is ever to come (ll. 277–281).
Beowulf's purpose, of course, is not just to advise Hrothgar on how to deal with Grendel—he fully intends to fight Grendel himself—but Beowulf is careful not to state his intent fully until he has a chance to meet directly with Hrothgar and understand the nature of Hrothgar's court. The dynamics of tribal culture require discretion when a relative stranger intrudes on a king's home territory.
The concept of family loyalty is introduced when Beowulf and his men are presented to Hrothgar, who immediately responds in a welcoming manner:
I knew him [Beowulf], when he was a boy; / his aged father was called Ecgtheow, / he whom Hrethel of the Geats at home gave / his only daughter; has his offspring now / boldly come here, sought a kind friend? (ll.372–376)
Hrothgar, well versed in tribal relationships, understands that Beowulf is not only a retainer of Hygelac, leader of the Geats, but he is also Hygelac's nephew, a particularly close bond in Scandinavian and Anglo-Saxon societies. More important, however, as we learn later in the poem, Hrothgar has a very close relationship with Beowulf's father, Ecgtheow.
Just after Beowulf formally introduces himself to Hrothgar, Hrothgar recounts the story of Beowulf's father, Ecgtheow, who "handed death to Heatholaf," a member of the Wilfing tribe, and the Geats cast Ecgtheow out of the tribe in order to avoid war. Ecgtheow then sought refuge with Hrothgar, who settles the dispute by paying the Wilfing tribe "wer-gild," that is, money to compensate them for the loss of a warrior. Beowulf's father, then, "swore me [Hrothgar] oaths." In other words, in a society that values loyalty, Beowulf's offer of help to Hrothgar is a natural part of his familial duty—on his father's behalf—to Hrothgar, and Hrothgar, having shielded Beowulf's father, senses a strong bond with Beowulf. Ecgtheow's "oaths" to Hrothgar are personified by Beowulf.
Consistent with the Scandinavian and Anglo-Saxon belief that personal glory is the most important achievement of a warrior, especially a warrior who wants to lead a tribe, Beowulf asks Hrothgar to give him permission to rid Hrothgar and his court of Grendel:
Chief of the Bright-Danes,...I will now/ask from you one favour:...I might alone purify Heorot (ll. 427–432).
Although Beowulf is well aware that Hrothgar and his warriors have not been able to conquer Grendel, he humbly and respectfully asks permission to help Hrothgar, an appropriate tone for a retainer to a king. Moreover, Beowulf, conscious of his path to personal glory, reminds Hrothgar that Grendel "in his rashness recks [has no use for] not of weapons":
I therefore disdain / to bear a sword or broad shield, / the yellow disk of war, but with my grip shall / fight with the fiend...there the one whom Death takes / will have to abide by the Captain's decision (ll. 336–441).
As he did with the fight against the sea-snakes, recounted earlier in the Breca episode, Beowulf seeks a "fair" fight with Grendel. The result, should Beowulf be victorious, yields more personal glory, which is the primary goal of a young warrior seeking to become his tribe's leader. These lines also contain a reference to "the Captain"—one of the poem's reminders that this story, even though its origins are pagan, is framed by the Anglo-Saxon's Christian belief system.
Beowulf's motives, then, are rooted in the primary imperatives of the Scandinavian and Anglo-Saxon warrior culture—loyalty to tribe and family, on one hand, and the accumulation of personal glory, on the other hand, both of which are necessary to live in world in which other tribes and natural (and supernatural) forces are constant enemies.