According to the Apocrypha, to protect Mary's virginity. By betrothing Mary to an elderly and righteous man as Joseph, Mary's virginity would be intact AND she would be provided for financially. This is not an unusual scheme, except to the 21st century American who is fixated on sex.
I find this hard to believe. What exactly was in this arrangement for Joseph? Let's see, he agrees to marry a child, and provide for her financially and for all her other needs. She, in return, agrees to not be a wife to him. Hmmm. Deal, or no deal? Oh, and why do you say this was not unusual? What, the practice of older men marrying much younger women for sex was just invented? :)
WHY DID JOSEPH WANT TO MARRY HER? If he wanted to take care of her financially, he sure could have without marrying her. Why the charade, why the lie?
Who is going to have sex with another person's wife? Not a normal person. You need to place yourself into the first century! SEX is not the only reason people get married. Honor was very important to the Meditteranean people. It still is, correct, Kosta? An older Joseph could provide protection to Mary's virginity. If you would have read the Infant Gospel of James, as was posted earlier by our Orthodox friends, this explanation would have become more clear.
Regards
The Protoevangelium says, he didn't, and beyond that we don't know.
8. [...] And when she was twelve years old there was held a council of the priests, saying: Behold, Mary has reached the age of twelve years in the temple of the Lord. What then shall we do with her, test perchance she defile the sanctuary of the Lord? And they said to the high priest: Thou standest by the altar of the Lord; go in, and pray concerning her; and whatever the Lord shall manifest unto thee, that also will we do. And the high priest went in, taking the robe with the twelve bells into the holy of holies; and he prayed concerning her. And behold an angel of the Lord stood by him, saying unto him: Zacharias, Zacharias, go out and assemble the widowers of the people, and let them bring each his rod; and to whomsoever the Lord shall show a sign, his wife shall she be. And the heralds went out through all the circuit of Judaea, and the trumpet of the Lord sounded, and all ran.9. And Joseph, throwing away his axe, went out to meet them; and when they had assembled, they went away to the high priest, taking with them their rods. And he, taking the rods of all of them, entered into the temple, and prayed; and having ended his prayer, he took the rods and came out, and gave them to them: but there was no sign in them, and Joseph took his rod last; and, behold, a dove came out of the rod, and flew upon Joseph's head. And the priest said to Joseph, Thou hast been chosen by lot to take into thy keeping the virgin of the Lord. But Joseph refused, saying: I have children, and I am an old man, and she is a young girl. I am afraid lest I become a laughing-stock to the sons of Israel. And the priest said to Joseph: Fear the Lord thy God, and remember what the Lord did to Dathan, and Abiram, and Korah; how the earth opened, and they were swallowed up on account of their contradiction. And now fear, O Joseph, lest the same things happen in thy house. And Joseph was afraid, and took her into his keeping.
Note that the priest is described to act as if the arranged marriage to whoever received the divine sign was the only way to go about it.