Hrothgar confidently puts his hopes (for safety and victory against Grendel) in Beowulf for a variety of reasons. One very superficial reason is that Beowulf looks the part of a big, strong, brave, experienced warrior. Early in the narrative, Beowulf is on his way to see Hrothgar. A guard stops Beowulf and wants to know Beowulf's purpose, and the guard also comments on what Beowulf looks like:
"Nor have I seen
A mightier man-at-arms on this earth
Than the one standing here: unless I am mistaken,
He is truly noble. This is no mere
Hanger-on in a hero’s armour."
This is the same image that Hrothgar will eventually see. Beowulf also has some very strong and heroic stories tied to his name. For example, we are told that Beowulf has the strength of thirty men.
"Who valuable gift-gems of the Geatmen carried
As peace-offering thither, that he thirty men’s grapple
Has in his hand, the hero-in-battle."
These things would help to give Hrothgar confidence in Beowulf as an experienced and strong warrior. The final thing that I think would give Hrothgar confidence is that Beowulf seeks him out. Beowulf wants to step into this fight. Beowulf isn't going into this fight because he is cornered and there is no other option—he is confident that he can defeat Grendel, and Hrothgar has to feel confident as well because of that. Beowulf isn't likely to step into a fight that is a guaranteed loss.