It's not only developing countries that benefit from increased female participation in the labour force. For Western nations, the increased proportion of women in the labour force was a core component of the rapid increased economic growth between the 1940's and today.
Low fertility is an interesting thing where it comes to development; for most countries it has quite dramatically declined as incomes have increased. The effects are likely bidirectional (increased economic growth -> lower fertility rate; lower fertility rate -> increased per capita income).
The drivers are complex and interrelated: high child mortality rates, low life expectancy, and high poverty are associated with high fertility. It's an unjust comparison to make, however leading theories share common features with K-r reproduction strategies in ecology (with increased survival / longevity, investing in a few highly educated/skilled children becomes feasible). Similarly, increased education / maternal health is highly correlated with economic development and child wellbeing. These are often highly compounding effects over time - a virtuous cycle in which increased economic freedoms, health, education, and opportunities (male and female - often females are relatively impoverished, so there's the potential for bigger immediate benefits) can lead to rapid development with the right institutional conditions.