People started living a lot longer. The 1.7 fertility index is only something very recent, historically China's had a very high birth rate (due to abysmal life expectancy).

In the 1950s the life expectancy in China was only in the low 40's. Then Mao came to power, and regardless of whatever his other fuck-ups were, he instituted sweeping healthcare reforms. In 1962 the life expectancy in China was 45; just 10 years later, in 1972, that had increased to 65. Meanwhile, during that time period, the birth rate peaked at over 6. Also, despite the single child policy, there were plenty of exceptions built in, so the birth rate didn't drop below 2 until 1994.

Source: World Bank historical demographic data

1.7 is the more recent fertility rate, it does not reflect fertility rate as a whole from the 1949-2015 era. In 1970 for example China had above 5 children per women on average. Fertility rate in China declined after the one child policy was implemented in 1979 due to both the policy itself and increasing living standards. However the country as a whole experienced massive population growth before 1979 while after 1979 increased economic prosperity brought longer lifespan for the Chinese population as a whole. In other words, the current increase in population is the result of earlier high birth rates as well as longer life expectancy for the current population level.

There're many reasons. 1949's communist ruling marks the end of many decades of war, so there's a natural rebound. And under communist ruling China experienced two vastly different period of fertility rate. During first 30 years (1949-1979), China government followed Soviet and encouraged reproduction. And Collective Farm system favors greatly families who have many children -- every one gets equal share of ration, whether you work better or worse. So in 1980, the population more than doubled from 1949, and median age was 20. Since then, China enforced one-child policy, and started economic reform, essentially removed collective farm system, so people has no economic benefit to give more birth, and instead has severe punishment.

Has anyone theorized what would have happened in China if the single child policy had never been implemented?