Beowulf, and the narrator, provide several reasons for Beowulf's choice, although not all of them are revealed at the same time. This may be due to the story being a compilation, or aggregate, with various authors adding their own elements to it over time, or because a staggered approach allows each piece of information to have more impact when it is considered by the reader in light of prior information.
The reason that Beowulf initially provides is that he will not use weapons because Grendel doesn't use any weapons either. This seems like it's a simple matter of making it a "fair fight" - although we must remind ourselves that Beowulf has been reputed to have a grip that is akin to the strength of 30 men, and Grendel appears more than capable of throwing men around like dolls, considering that he dragged 15 of them back to the marsh in a single raid. By fighting Grendel without weapons, Beowulf also leaves no room for anyone to argue that it was his armor or his sword, rather than his body and mind, that defeated the monster. In a sense, Beowulf will prove himself the better monster.
Another reason for scorning weapons is the reputation this will bring to Hygelac. It is also mentioned during the fight with Grendel's mother that Beowulf casts his sword aside and seizes her with his hands, and to "trust in one's strength" is the best and only way to win honor for one's lord in battle. As demonstrated by his generous gift-giving to Hygelac and Hygd upon his return, Beowulf recognizes the value of keeping one's lord in high esteem and representing him well, for this will earn him trust and reputation.
Finally, it is mentioned during the fight with the dragon that Beowulf had never had luck with swords; his strength was so great that he tended to simply break them. It is interesting that this is not mentioned earlier, and by suggesting that Beowulf's use of his hands is a necessity might imply that he is in some ways beyond the taming influence of civilization and refined tools.
The epic of Beowulf illustrates the characteristics of the epic hero through the battles fought by Beowulf. Beowulf knows that Grendel uses no weapons, only his bear hands. As a true hero, Beowulf decides to do the same.
This equality is based upon the ideas of aristeia and superari a superiore. Aristeia refers to the nobility of a person, and superari a superiore refers to success over a superior foe. If Beowulf would have used a weapon over Grendel, the weapon would have provided Beowulf an edge (making Beowulf the superior fighter). In order to equal the field, Beowulf did not use any weapons. This speaks to his nobility and honor as a true epic hero.