Why does Beowulf slay Grendel?
Beowulf doesn't really announce a specific reason for his intention to slay Grendel. The tone of the poem, and the manner in which the author regards Grendel (calling him hellspawn and a descendant of Cain, who killed his own brother and was damned to wander the earth forever), makes it clear that the author means for us to interpret Grendel as a near-supernatural, purely evil entity with no redeeming qualities nor any possibility of negotiation or peace. Just as good and evil are clearly delineated in the Biblical sources the author draws upon, so too is Grendel clearly evil and therefore deserving of retribution and death, according to this morality. Beowulf doesn't need to explain why he kills Grendel because, as far as the author and the characters in the context of the poem are concerned, it is self-evident that Grendel must be slain in order to stop the attacks on Heorot.
There is a small concession to diplomacy on lines 154 to 160, where it is mentioned that Grendel has no intention of stopping his attacks, and nobody expected him to pay the money normally demanded of a murderer to stave off retribution from the murdered person's kinsmen.
Beowulf does mention, beginning at line 960, that he wishes Hrothgar could have seen Grendel rather than just his disembodied arm; Beowulf wanted to grapple Grendel into submission and then kill him, but by chance, Grendel was too slippery for Beowulf to completely subdue. Beowulf states that Grendel basically "traded his arm for his life", but he knows the wound was mortal and Grendel is likely dead already.