I'm not sure if the word "line" in this paragraph means "words"?
Thus, after Grendel’s death, King Hrothgar speaks sanguinely of the future, which the audience knows will end with the destruction of his line and the burning of Heorot.
It means his "bloodline". (sense 17) It means that all Hrothgar's descendants (sons, grandsons etc) will be killed, and the kingdom will pass from his family (to Beowulf).