Access to better healthcare lowers the infant mortality rate. That than leads to couples having fewer children since they don't need to hedge their bets as much (ie having a more offspring expecting several to die). Combine that with easier access to contraceptives and you have some major reasons why birth rates decrease in developing countries which have the money to spend on healthcare.

In developed countries you may also have factors like people focussing on careers, valuing independence, more choice in relationships (i.e. divorce/separation being more acceptable) etc.