TFR (total fertility rate) plummets because it used to be necessary that it was way higher when the child mortality rate was higher - and indeed a decreased in fertility rate follows a decrease in child mortality with a few years of delay.

Having a child used to be a random experiment and you needed to make sure that one makes it through to care for you when you will be old or sick. Pensions, health and welfare systems and much better child mortality rates make it unnecessary to have many children. A replacement fertility rate of slightly over 2.0-2.1 is/would be fine in developed countries.