If you have a large number of young people in a country with a life expectancy in the 80s, almost all of them aren't going to die until their 70s.
But if they have an average of 1 child each (and by their 40s, this is pretty locked in), there is a long-term population decline locked in.
An easy way to calculate the fertility rate is to work out what percentage of women at each age have a child that year, work out a woman's death rate, and then integrate.
I'll make a toy country. In this country, people live exactly 80 years, then drop dead.
Women have a 1.5 children at age 40 (exactly), starting 40 years ago.
Prior to 40 years ago, women had 4.0 children at age 40 (exactly).
120 years ago, start out with 80,000 women evenly of every age (1000 < 1 etc).
So there are 4000 kids born, 2000 of them women. This happens for the next 40 years, at which point there are 8000 kids born, 4000 of them women. Another 40 years pass with that birth rate.
We are now 40 years ago, and fertility plummets.
Now, death rates are based on *how many people where born 80 years ago). This is 4000 people (2000 of them women).
There are 8000 women turning 40. They each have 1.5 children, having 12000 children, 6000 of them women.
Meanwhile, only 4000 old people die. We have fertility below replacement (1.5 children per woman per lifetime), but birth are greater than deaths!
This continues for the next 40 years. Each year, the 8000 women turning 40 all have 1.5 children (12000 total), 6000 of them are women, and 4000 80 year olds (2000 of them women) die.
We now hit year 0. Now, only 6000 women turn 40; so they have 9000 children, 4500 of them women. 80 years ago 8000 (4000 of them women) kids where born, so 8000 80 year olds die.
The gap between births and deaths remain, despite 40 years of low fertility.
40 more years pass at this rate - we are at year +40. Now only 4500 women turn 40, giving birth to 6750 children, 3375 of them women. Meanwhile 12000 people turn 80 and die.
It isn't until 80 years after fertility dropped that births finally fell behind deaths, despite the fact that nobody lives past 80.
Why does this happen?
Birth rates are based off of the birth rates of women over the fertile ages, very roughly 18 to 43, times the fertility per year.
Fertility rate is the integration of the current fertility rate for each year.
Death rates is mostly based off of how many 70+ people there are in the country; death rates are pretty low prior to 70.
So one tracks births 30ish years ago, the other tracks births 70ish years ago. So long as birth rates are going up 30-70 years ago sufficiently fast (to make up for the sup-1 fertility rate) this means births are greater than deaths.
The toy example made this really sharp, but the effect remains with less sharp situations.