As with all liberal arguments, there is no moral, ethical, logical foundation; so points are advanced on emotion alone.
On the contrary, it is the conservative position that is advanced on emotion, tugging at our natural emotional response to children. As conservatives often argue, no person has a right to the services of another. Healthcare, for example, is not a right because it requires a doctor’s services. The doctor is free to charge for his work or to refuse service. In similar fashion, it can be argued that you do not have a right to be born, because it requires work on part of your mother. Later, you would also become a financial burden on her when she will be required to feed you and pay for your education and medical care. Naturally, most of us welcome these things because we want to have children and because we love them, but it would not be moral to force a woman to give birth to and raise a child she does not want.
Abortion removes the child from the womb; it does not have to kill it. Death just happens because the child can not survive independently. Does this automatically mean that the child has a claim on the mother’s body, that she is somehow morally obligated to carry him to term because he needs it? The socialist argument is yes; you must always work for those who need you. The conservative argument is yes, because God creates all life in his image and all life is therefore sacred and must be preserved at all cost, even if it means forced labor. The liberal (classical liberal, anyway) argument is that forced labor of any kind is slavery and is never justified, so it is preferrable to allow the child to die than to force its care upon the mother.
Each of these views is perfectly logical and follows naturally from the core moral principles. Those would be altruism for socialists, the sanctity of life for christians, and the rejection of forced servitude for the rest.