Good question. There are many reasons. Revenge is one. He must destroy here because of her attempt to seek revenge on Beowulf and his men for killing her son. This is similar to the blood feuds the Anglo-Saxons fought.
Glory and to cement his reputation are also factors. Remember, Beowulf\'s people, the Celts and later the Anglo-Saxons before Augustine converted King Ethelbert to Christianity, didn\'t have a concept of afterlife. The only way to live on was to perform heroic deeds that would be recorded in song and poem by the bards and recited after you were dead. By not only killing the monster Grendel but also killing his mother, Beowulf\'s reputation would be even more incredible and sure to be retold.
Bravery is another factor. Remember the heroic code to which Beowulf adheres - loyalty, bravery, and generosity - are embodied in all of his battles. He is being loyal to Hrothgar by sailing to his land to battle these monsters. If he were to leave, he would not fulfill his obligation. He is also being brave by seeking her out and trying to kill her. Finally, if he destroys her, his men can share her treasure and weapons. All of these are vital to the Celtic and Anglo-Saxon ways of life.
All of these are great possible answers. At this point in the poem, however, Beowulf is going mainly for justice/revenge. The men had celebrated heartily after Grendel had been killed and his arm is hung in the rafters as a sort of trophy. They let their guard down and once again return to the Hall of Herot as a safe haven and place to lie down to sleep without worries. Grendel's mother comes without warning and not only takes back her son's arm, but kills a few men (among them Hrothgrar's best buddy). Beowulf takes it upon himself to seek justice for harm done and revenge for Hrothgar (as he owes this to the long-family friend for once helping out Beowulf's father).