As all of the previous posts noted, the act of hanging Grendel's arm over the rafter is a way for the poet to show Beowulf's prowess as a warrior. Never one to miss a chance to boast, Beowulf opted to create a showy visual of his feat.
But the poet also uses Beowulf's action with Grendel's arm to create something called "unity" in the plot. Writers create unity when they give their story a sense of completeness, a feeling that all of the events and key ideas in the story are thematically related and necessary to make the story complete. When Grendel's mother, furious over the killing of her son, bursts into Heorot looking for revenge, she does two things: She kills Hrothgar's best friend, and she takes her son's arm with her. She's a pretty smart monster and she knows how to hurt both Hrothgar and Beowulf. By having Grendel's mother remove Beowulf's symbol of victory, the poet has reminded us of Beowulf's earlier action and given the story a sense of completeness, of unity, that it wouldn't have otherwise. Most good stories, especially if they are not short stories, have multiple incidents that help build unity.