How does Beowulf prove to Hrothgar with assurance that he is capable of ridding the Danes of Grendel?
One answer is that Beowulf proves he can take care of Grendel when he and Grendel actually fight. As a result of the fight, Beowulf is able to tear Grendel's arm off:
The awful monster had lived to feel pain in his body, a huge wound in his shoulder was exposed, his sinews sprang apart, his bone-locks broke.
Following this battle, Hrothgar praises Beowulf, saying "Now, Beowulf, best of men, in my heart I will love you as a son: keep well this new kinship."
However, it is more accurate to say that Beowulf assures Hrothgar of his abilities before any of this. Prior to the battle with Grendel, Beowulf responds to Unferth who criticizes his (Beowulf's) abilities. Unferth claims that tales of Beowulf's heroism are exaggerations. He points out that Beowulf lost a swimming contest with Breca. Beowulf responds, adding that he was the stronger swimmer and had to contend with nine sea-monsters during the contest. Beowful adds that there have been no such feats committed by those of Heorot (Unerth included): "I have not heard say of you any such hard matching of might, such sword-terror. Breca never yet in the games of war -- neither he nor you--achieved so bold a deed with bright swords . . ."
This refutation of Unferth's criticisms seems to have assured Hrothgar.
Never before, since I could raise hand and shield, have I entrusted to any man the great hall of the Danes, except now to you.
Beowulf's tales of heroism are hard to believe, but at this point, Hrothgar is assured (or wants to believe so badly that he becomes assured) that Beowulf will live up to his legendary status.