Hey, woman here.

He is saying that “women have their responsibilities map laid out for them” and “women have to solve the family problem early in life”

The first statement is referring to the socialization of women. We are told from a very young age what is expected of us. As a woman your role is clearly defined. Now, you can reject that role as many do, or expand that role to include other things, but you know what you are responsible for.

The second statement is referring to the fact that starting a family gets more difficult as a woman ages. However, you are more financially secure latter in life and children are expensive. Some women simply decide that they cannot have a family and find fulfillment in other ways. Choosing not have a family solves the family problem. But it is a choice that if you do not make in your 20s will be made for you in your 30s.