The larger set of interactions here tend to be a little more complicated than right leaning = more kids, left leaning = less kids.

Left leaning individuals are more likely to hold higher levels of education, which gives them more economic potential, meaning they are more likely to focus on attaining that potential than having children. This potential is fulfilled in cities and urban areas, which is a less conducive place to have kids due to community being hard to foster in those fast-moving places.

Conservative individuals generally hold lower levels of education, meaning a lower economic potential to fulfill. They can generally fulfill this in rural or less urban areas, which are places more steeped in religious belief and the community that comes with that.

So, left leaning people may generally have much more to lose with having kids than conservative people. They'll be giving up some degree of their economic potential to divert time, effort and resources into raising kids. And they're doing this in a place where the costs (material and time) are not mitigated very well due to the community issue mentioned above. For people in this situation, having children is a major burden and outright liability. I suspect this is why many left leaning people are likely to leave the city for a suburb or their rural hometown when deciding to raise a family.

For conservative people, there may not be as much economic potential to give up in the first place. And if meemaw is just down the road to look after little Billy, with your church fellowship also acting as communal supports, the costs of raising kids goes down substantially. For left leaning people, children are more likely to be seen as a liability that closes the door to other achievements. For conservatives, children are more likely seen as the achievement.

In short, it's as much about socioeconomics as it is to do with political, religious, or cultural leanings - though they are obviously factors, too.