Both can be true, that we’re experiencing record low birth rates globally and that the global population is still increasing at the moment.

How?

1. While birth rates in many countries have fallen below replacement rate, it’s still not zero, which means people are still having babies.
2. Due to advances in health science, the death rate has fallen.

These two factors, especially decades earlier, mean that population hasn’t yet fallen. However:

3. Non-existent humans will not produce babies.
4. The older the population is on the average, the higher the death rate will be.

This means that if I don’t produce offspring, my non-existent offspring will not produce babies. The less babies are produced, the older the population would be, and the higher the death rate will be. If current trends continue, the death rate will overtake the birth rate, and the population will shrink.

Outside of a worldwide disaster that kills off people of child-bearing age, population will still rise before it levels off and then fall off as more and more people find less and less appealing to raise children. This is just a consequence of us humans not dying immediately after childbirth, and us humans as a whole making offspring at a certain age (say, 20 years old). These two factors explain the lag between childbirth figures and population growth.