There's a very simple reason. The stakes are tremendous on both sides.
On the one hand is human life. The pro-life side is correct that there is human life from the moment of conception. In fact, human genetic material -- eggs and sperm -- are also "alive." All human life is interconnected. There is an unbroken chain of life going back to the first "homo sapiens" tens of thousands of years ago, back through the great apes, the primates, and so on to the first genesis of life on earth nearly four billion years ago. Life is special. Life is precious. It is wrong to destroy life.
On the other hand is a woman's right to control her body and her reproductive choices. Human history is rife with men's desire to own and possess women. Read the Bible! Families sold their daughters into slavery, concubinage, and marriage, all of which were similar. Marriage itself was little more than legalized rape. Laws limiting the rights of married women ("Coverture Acts") were still on the books long after I was married.
It is only in the most recent past that women have emerged as having equal rights to men. Women gained the right to vote in 1920, two years after my mother was born. They gained the right to use contraception (if married) in 1965. They gained protection under the Equal Protection Clause in 1971. They gained the right to use contraction if single in 1972. The gained the right to terminate an unwanted pregnancy in its early stages in 1973.
It's not as though our society has a long history of treating women equally. Good heavens, women only gained the right to be admitted to a prestigious state university on the same basis as men in the year 1996!
A woman's body belongs to herself and her alone. Under our Constitution she now has the right to choose whether to have sex, who to have sex with, and whether to have children. It is wrong to interfere with those choices.
The decision of the Supreme Court of the United States in Roe v. Wade was a compromise between these fundamental values of life and choice. The Court ruled that women have a constitutional right to terminate a pregnancy prior to viability, that is, up until the point that a fetus can live outside the womb, about 24 weeks into the pregnancy. Until then the woman has the right to determine whether to bring forth this new life; after then the choice is no longer hers, but may be controlled by law.
People who think that there is an easy answer to the question of abortion fail to consider the fundamental values that are at stake on both sides of the constitutional equation.