If you want the ideological underpinnings of these arguments, there are a few.

The biggest one always boils down to the assertion that Government Is Paying For The Abortions. And that this is bad because it creates an incentive for young people to go out and have unmarried sex.

But there's a second, arguably more insidious argument that you see crop up on the libertarian-right, which asserts the idea of "negative rights". These are obligations of inaction that a state can impose on an individual or group, on the grounds that is makes others more free. In this case, the argument is that the fetus has a negative right with respect to the mother. Once you're pregnant, you no longer have rights to your own body because another person is occupying it. This reasoning stems from the claim that fathers have an equal share in the property that is the child. This community property needs to be protected from a woman who wants to discharge her obligations prematurely at a loss to the man.

Pregnancy becomes a kind of debt that an impregnated woman owes to the impregnating man (and, by extension, the man's family who also gets some degree of claim to the fetus). The woman is in debt bondage until the pregnancy is over. And the state - which libertarian ideology asserts should exist only to enforce property rights - has an obligation to obstruct the woman from evading payment of what is owed.

By contrast, all those state liabilities - health care, education, social services for children - are "positive rights". Libertarians assert that these rights are actually an unlawful infringement on one human by another, because they must be paid for out of a communal set of resources. The mother cannot demand any recompense for her maternal state. The child cannot demand any basic standard of living as a minor. These all have to come from another adult (in a libertarian patriarchy, that means the father) and that once again infringes on his property.

It really does just boil down to the idea of Communism Bad.