does not use weapons, and Beowulf wants to fight in the same way. Another, more personal reason he mentions is that his king, Higlac, would be disappointed in him if he were to fight this monster in any other way. Beowulf is a mighty warrior and knows his own strength. He has boasted about ridding the earth of giants and the seas of monsters, so he does not feel the need to fight with weapons.

In the Anglo-Saxon epic poem, Beowulf, the protagonist Beowulf insists on battling Grendel with his bare hands, instead of using a weapon, because, he says, Grendel doesn't use one. 

The sense of honor Beowulf displays here is one of the main themes of the poem.  Beowulf sees using a weapon against a beast that uses no weapon as dishonorable.  He will fight Grendel on equal terms.  It is the honorable thing to do.  Grendel is charmed and cannot be cut by swords, but Beowulf does not know this as far as a reader can tell, and this is not why, according to his own words, he refuses to use a sword. 

As he says:

...I have heard,

Too, that the monster's scorn of men

Is so great that he needs no weapons and fears none.

Nor will I.  My lord Higlac

Might think less of me if I let my sword

Go where my feet were afraid to, if I hid

Behind some broad linden shield:   my hands

Alone shall fight for me, struggle for life

Against the monster.  God must decide

Who will be given to death's cold grip.  (261-270)

Of course, no one else mentions this, and others have fought Grendel with swords.  But this is one of the character traits that makes Beowulf different and special.  His sense of honor is even greater than that of the rest of his society.  That's why he's an epic hero.  He could also give a great speech!

I am not sure how Beowulf knows this, but the reason that he does not use weapons when fighting Grendel is that the monster has some sort of magical protection.  The monster has managed to do something that prevents any sort of edged weapon from doing him harm.

So, instead of trying to kill the monster with a sword or spear or anything like that, Beowulf fights him hand to hand and tears his arm off.

You can find this in Chapter 12 if you need to be able to cite proof of this answer.

There are at least three reasons for Beowulf's decision to face Grendel without a weapon--one based on circumstance; one based on what the poet tells us about a weapon's effectiveness; and one based on Beowulf's desire for fame.

First, as Grendel attacks Heorot (with Beowulf and his men asleep but anticipating an attack), Grendel immediately seizes one of Beowulf's men, rips him to shreds, and devours him "feet and hands" (l. 45).  Next, Grendel moves to an adjacent warrior:

Nearer he stepped, and/grasped by the hand the great-hearted fighter [Beowulf] where he lay; Beowulf parried with his hand; he took hold on him quickly. . . . (ll. 45-48)

Grendel has fatefully grabbed the one warrior who is capable of defeating him, and using a defensive move based on circumstance, Beowulf, who probably has no room to maneuver with a weapon, first avoids Grendel's grasp and then grabs Grendel himself: "he took hold on him quickly/with fierce purpose and pinned his arm." (ll.48-49)

Second, as the struggle between Beowulf and Grendel continues, the poet tells us that Beowulf's warriors, in an effort to help protect him from Grendel, drew their weapons.  Unfortunately, according to the poet, Beowulf's warriors

. . . didn't realize that no war-blades . . . could even scathe that sinful wrecker;/for he'd case spells against all/edged weapons whatever. (ll.801-805)

We learn here that one of the reasons for Grendel's success over the years is his invulnerability to the weapons of mankind.  The poet subtly implies that only a man of Beowulf's strength and stature, using only his physical strength, has any hope of defeating Grendel.

The third, and perhaps most important reason, is cultural.  Beowulf, one of whose goals is to seek personal glory, understands that defeating Grendel with only his physical prowess is crucial to achieving the greatest amount of fame in this contest.  Using a weapon--assuming a sword or axe would be effective--to defeat Grendel would not lead to the same level of fame because Beowulf's detractors (Unferth, for example) could claim that Beowulf's success was the result of using a weapon against an unarmed (no pun intended) monster.  In this contest, the odds are equalized by the fact that neither fighter is armed with a weapon.  Beowulf's fame in the battle, therefore, is enhanced by his decision not to use a weapon.  And fame is the goal.

There are several reasons why Beowulf wants to fight Grendel without the use of a sword. His famous "boasts" of bravery in battle is one reason-he wants people to talk about it like they have talked about his other conquests. For instance, he loves the fact that Wiglaf has talked about his fight with the dragon under water and the fact that his speech backfires on him. He also wants to prove it to himself that he can do it. I think there is part of him that wants to fight Grendel fairly. Grendel has no weapons, so he wants to meet him on his terms, as seen below:

"Of force in fight no feebler I count me,in grim war-deeds, than Grendel deems him.Not with the sword, then, to sleep of deathhis life will I give, though it lie in my power.No skill is his to strike against me,my shield to hew though he hardy be,bold in battle; we both, this night,shall spurn the sword, if he seek me here,unweaponed, for war. Let wisest God,sacred Lord, on which side soeverdoom decree as he deemeth right."

He wants to battle Grendel "unweaponed" because Grendel cannot help but "spurn the sword." The final reason is illustrated in the last line of this passage: "Let wisest God, sacred Lord, on which side soever doom decree as he deemeth right." Beowulf trusts on God to help the righteous winner win.