2 of 4 Added CCC 2376 with link.

I'm not a Catholic and by no means an expert, but my understanding is that the Catholic argument against artificial means of conceiving is the same as the argument against euthenasia: that it's tantamount to "playing God." Matters of life and death are in God's hands; the Catholic position is that we aren't supposed to take them into our own hands.

This is the relevant paragraph from the Catechism of the Catholic Church on the issue:

CCC 2376 Techniques that entail the dissociation of husband and wife, by the intrusion of a person other than the couple (donation of sperm or ovum, surrogate uterus), are gravely immoral. These techniques (heterologous artificial insemination and fertilization) infringe the child's right to be born of a father and mother known to him and bound to each other by marriage. They betray the spouses' "right to become a father and a mother only through each other."1

167 CDF, Donum vitae II,1.