DEUT1441 If a man dies without children, his brother performs yibum (marries the widow). A woman, when she marries, becomes like a limb of her husband. If he dies childless, her connection to his family does not dissolve. So that he gains a foothold in the physical world and will not be forgotten from it, his brother must marry her, and if a child is born to them it will be considered the offspring of the deceased, because the mother still is considered one of his limbs. The child will cause people to remember the deceased, who will have left someone behind him in this world to carry forth his name and serve Hashem. Another reason is that the man who died leaves nothing behind in this world except for his wife, who is like his own flesh. Therefore, Hashem desires that a child be born to him by means of a union between his widow and one of his brothers, for a brother, too, is like part of one's own flesh. The child of such parents is truly like a continuation of the deceased himself. The child will serve Hashem in this world as if the deceased is still here serving Him. Thereby, the man who died childless will, in fact, have offspring in this world. The mitzvos of “heir” and “replacement” shall bring him merit in the World to Come, for the good deeds of a son bring merit to his father.