He is speaking in Old English (sometimes called the Anglo-Saxon language), from which modern English is derived. It is closer to its German roots. Here are his lines, translated (some of Grendel's mother's words are in Old English as well):
(In Grendel's Lair, after his first attack)
GRENDEL'S MOTHER: Grendel, what have you done? What have you done, Grendel?
GRENDEL'S MOTHER: Fish and wolf and bear, and sheep or two, but not men.
GRENDEL: Men? They are small.
GRENDEL'S MOTHER: Men, Grendel. They have slain so many of our kind.
GRENDEL: The men screamed! The men bellowed and screamed! The men hurt me, hurt my ear.
GRENDEL'S MOTHER: Was Hrothgar there?
GRENDEL: I did not hurt him, I did not hurt him, no.
GRENDEL'S MOTHER: Good. Good boy, and tender.
(Beowulf and Grendel battle in the mead hall)
BEOWULF: Your bloodletting days are finished, demon.
GRENDEL: I am not a demon.
BEOWULF: It speaks. It speaks!
GRENDEL: What are you?
BEOWULF: I am ripper, tearer, slasher, gouger....
(Grendel returns to his lair)
GRENDEL'S MOTHER: Grendel. My son. My poor son.
GRENDEL: Mother. They hurt me, Mother.
GRENDEL'S MOTHER: Sleep now, my son.
GRENDEL: He murdered me, Mother.
GRENDEL'S MOTHER. Who murdered you, my son?
GRENDEL: He ripped off my arm.
GRENDEL'S MOTHER: He will pay, my darling. Who was the man?
GRENDEL: He was so strong. So strong.
GRENDEL'S MOTHER: Who was the man?
GRENDEL: His name was Beowulf.
GRENDEL'S MOTHER: Beowulf.