Actually, Beowulf as hero does not tell these stories; they are told in third-person in the voice of the narrator. The epic itself is an oral history of Beowulf's great feats--his life, his death, and his burial.
The detail in which his battles with Grendel and Grendel's mother are recounted serves several purposes. It makes the stories dramatic, exciting, and compelling; these are tales of danger and violence, with strong supernatural elements. The specific details make them come alive for the audience, creating dramatic images that the audience can visualize. Beowulf not only tears off Grendel's arm; he hangs it from the rafters in the mead hall--quite a dramatic image there.
More importantly, by recounting Beowulf's battles in such vivid detail, the heroic aspects of his character are established and emphasized. His strength, courage, and determination are heroic as he battles larger-than-life enemies that no one else can slay.
The details that establish the strength and evil nature of the monsters emphasize the heroic elements in Beowulf's character, as well as his personal goodness. He challenges the evil monsters at the risk of his own life for the benefit of a society not his own. Also, details establish that Beowulf possesses supernatural abilities, sinking through the sea for hours upon hours before even reaching the underwater den of Grendel's mother. The many specific details of Beowulf's battles are essential in developing him as the epic hero of the story.
In the epic poem, Beowulf, I believe there are probably several reasons for the extensive detail used in Sections 29-30.
First, I believe that the storytelling by the character of Beowulf, in the details he chooses to include (keeping in mind that Beowulf is the narrator here), are provided to give a clearer picture of the hero Beowulf is. As was the custom, anything that a warrior received as a reward was then returned to his lord or king to honor him. Beowulf is certainly a great warrior, but he also loves and honors Hygelac, his feudal lord.
Wikipedia.com points to this tradition in the following passage:
In addition, Beowulf’s defeat of Grendel prompts the Danish king to bestow upon him many gifts consisting of weapons; this further emphasizes the importance of weaponry to such a society. Beowulf then passes on his rewards to his king Hygelac, thereby establishing his obligation to his king.
The second reason I believe the poet recounting Beowulf's adventures goes into so much detail is for the sake of the actual process of telling the story.
For many years, before people could read and write (and for the few who could, only the Church primarily had written records), stories were passed down through the oral tradition, or by word-of-mouth. These stories were passed down usually through a scop. A scop would often times be the keeper of the stories of a tribe or culture: this would be his or her full-time job (and a scop could be a man or woman, and pass the job and knowledge to son or daughter), to pass along information about kings, rulers, battles, heroes, etc.
Sometimes a town would have its own scop. Sometimes a scop would travel from place to place, as not all villages could support a scop. The scop would stop at a castle or hall: the more stories he/she had to tell, often to the accompaniment of a drum beat, and the more songs he could sing, the longer he would be welcomed—housed and fed.
The sharing of these details would provide for a longer story, would show Beowulf to be a great and honorable warrior, and would also engage the imagination of an audience that had little relief from the tedious nature of each day. One worked from daybreak to sunset. There were no other forms of entertainment after dark: not even reading. The presence of a scop would have been a great treat.
And as is still the case today, the story of someone being showered with rewards for being heroic is appreciated by those listening: everyone would like to imagine being in such a position. A good story was valuable then, as it still is today.