How does Beowulf trap and kill grendel?

He rips off his arm and places it in Herot. Grendel runs home and laters dies in his lair but his mother goes back to Herot to kill and gets away until Beowulf goes to her lair and cuts off her head. He does find Grendel, however he is already dead by that point.

Answer:

As to the physical how-

In the Words of the actual poem (Seamus Heanley Translation), starting at line 815

" ... a tremendous wound appeared on his (Gendel's) shoulder. Sinews split and the bone lappings burst ... Grendel was driven under the fen-banks fatally hurt. to his desolate lair" After the battle with Grendel's mother Beowulf finds Grendel's body in the cave "... in its resting place ..." and beheads it. (Line 1580)

The symbolic how is diferent. Both Grenel and Beowulfrepresent the culture of the Vikings. Grendel the old ways with the worship of Odin and the other Aesir, Beowulf the new wave of Christianity coming to the land. Both are nearly invulnerable but Beowulf defeats Grendel as the new culture will overtake the old.