Beowulf hung Grendel's arm, or claw, on the rafters as a symbol of victory, both to secure his own legacy and to improve morale among his men, according to Cliffs Notes. The story of Beowulf is full of trophies, as would have been common among warriors at that time. Beowulf had hoped to present Grendel's body to the king but settled for the arm.
The hanging of Grendel's arm enraged his mother. Consequently, she attacked Heorot, taking the arm back. She also killed one of Beowulf's men in the process. Beowulf later tracks her back to her lair, ultimately killing her and taking Grendel's head as a bigger trophy of his victories over the monsters.