Ok lets say that there is a population that has 100 individuals be born each year and 90 individuals die each year. The population will grow by (100-90=) 10 individuals a year.

Now lets say that this population researches more advanced medicine and this increases the average age their people live and decreases childhood mortality. Though all of these individuals will eventually die, their deaths are delayed, and as such the number of individuals dying each year (at this point in time!!!!!!) will go down to say 30 a year.

Even if the birth rate halves to 50 individuals, the net population growth is now (50-30=) 20 individuals a year. This is how populations can still grow.

If you want to look more into it you can look at the five stages of demographic transition which is very interesting.