Cremation with one's possessions is given as Odin's law in the Ynglinga Saga. The reasoning given is that cremation with one's possessions allow them to bring whatever was burned with them into Valhalla:

Odin established the same law in his land that had been in force in Asaland. Thus he established by law that all dead men should be burned, and their belongings laid with them upon the pile, and the ashes be cast into the sea or buried in the earth. Thus, said he, every one will come to Valhalla with the riches he had with him upon the pile; and he would also enjoy whatever he himself had buried in the earth.

An early account of such a ship burial comes from an account of Ibn Fadlan (a 10th century Arab traveler) among the Rus. It is similar in that the final burial involves the deceased placed in a ship and then burned. The ship, in this case, was aground and was not lit by arrows, however.

The reason for burning the ship is not specifically addressed, necessarily, but the implication seems to be that it is meant to accompany him to the afterlife along with the rest of his possessions. The man is cremated with numerous other possessions as well, including food, drink, an instrument, fine clothes, weapons, animals, and a slave girl. These are to be with him in the afterlife, as made clear as they instruct the to-be-sacrificed slave girl to relay a message to her deceased lord:

The girl slave who wished to be killed went here and there and into each of their tents, and the master of each tent had sexual intercourse with her and said, "Tell your lord I have done this out of love for him."

The Rus also give a brief explanation of why cremation is chosen, as opposed to burial: