I've noticed that government benefits to families are often reduced if there is some other means of support. The following applies to the only state I know this stuff about, Washington.
For example, if a woman is living alone and gets assistance with her rent, if she announces that she has a room-mate, the government will give her less for rent. They assume the room-mate will pay some. Similarly, welfare for single mothers typically has policies that encourage her to marry or locate the father to force him to pay child support - which results in a deduction from the welfare. The government wants people to support each other.
The state also likes to assign custody of the kids to the mother in the event of divorce. I suspect this may be because she's less likely to abandon them than the father due to a belief in a stronger maternal attachment than a paternal one (although whether this difference actually exists is debatable, I assume they operate under the misconception anyway).
A single mother has difficulty both working and raising the kids. She needs to pay for child care or else have abnormal work hours, both of which are straining to the budget. She also must support multiple people, and growing kids go through clothes and food very quickly. The other single parent, the father, likely has better earning potential and also fewer demands on his income.
So it makes sense that, if a family breaks up and the kids stay with the mother, the mother will need assistance. This typically takes the form of child support, but if the child support isn't forthcoming the state steps in to make sure these kids don't starve to death.
This ignores the issues of same-sex parents (who may be equally able to demand custody and have equal wage-earning potential) or nontraditional gender roles in opposite-sex parents (the woman is the breadwinner, male stays at home to raise the kids - upon divorce, if she gets the kids, he has much lower than expected earning power because he took the traditionally female role and did not develop his career skills useful outside the home. This may also be a problem with same-sex couples), and multi-parent households (if you have a woman and two men, and multiple-marriage isn't legal, will the state impose a common-law marriage? Which man will it determine is the husband? What if she has custody of two children, one from each man? What if neither man is the father of these children? Will the state split the child support responsibility between the two, creating a de jure three-parent relationship?).
Anyway, all this is just describing what I think is the modern rationale behind the government's attempts to get people to marry and stay married, and failing that to get the divorced parents to continue sharing resources: because it reduces the government's burden in supporting them.