The writing of Beowulf has been problematic for many readers. That said, modern translations sometimes help readers cope with the words, phrases, and meaning of the story. Sometimes, by rephrasing pieces of a text, one can come to understand it better (although it does take a little longer).
A paraphrase is a restating of the original in other words. Therefore, the passage in question will be paralleled with the paraphrased meaning following in bold (line by line).
So dutiful thanes
(Then the motivated men)
in liege to their lord mourn him with lays
(in obligation to their leader showed sorrow with song)
praising his peerless prowess in battle
(telling of his matchless skill in conflicts)
as it is fitting when life leaves the flesh.
(which is expected when one dies
Heavy-hearted his hearth-companions
(Saddened his fellow warriors)
grieved for Beowulf, great among kings,
(felt the loss of Beowulf, the best of all kings)
mild in his mien, most gentle of men,
(gentle in his manner, the most tender of men)
kindest to kinfolk yet keenest for fame.
(friendly to family although eager for glory)