I need clarification, please. Why does the "Jewishness" of a child come through its mother? Are not the Jews descended from Abraham, a man? My father is Jewish, but my mother isn't. Why aren't I a Jew?
The legal dimension is that matrilineal descent is something that dates back to Sinai and the giving of the Torah. The Mishna and the Talmud are full of cases that prove that one's "Jewish genes" are transferred from the mother. The Torah itself also alludes to this point, see Leviticus 24:11. The Mishna in Kiddushin 66b states that if a child's mother is not Jewish, then the child is "like her," (i.e., not Jewish). This Halacha is codified in the Shulchan Aruch, Even HaEzer 8:5, without mention of any dissenting opinion. No source in the Torah teaches otherwise, and this question has never been raised in any classical Halachic text. It is an obvious and accepted axiom given to us at Sinai.
The philosophical dimension is that Tribal Affiliation is passed down from the father to his children whereas Jewish identity is passed from the mother to her children. The Rabbis explain that Tribal identity is external and one's Jewishness is internal. The male is a much more "external being" and the female is much more "internal", as can be seen clearly through the pregnancy. The male donates a part of him that leaves his body (i.e. is external) whereas the female has the fetus growing inside of her and she nurtures the child through pregnancy as if the fetus is a part of her. Because one's relationship with God is extremely personal and internal it is the mother who defines whether the child is Jewish or not.