If a population grows........

If a population grows at a rate of r(t) people per year, what does
∫ r'(t) from t=0 to 10 represent?

Answer: It represents the total amount of people after 10 years
Why is this the answer?

That is not the right answer. The answer is that the equation represents the number of people that were born during those ten years, not the total population. If time = zero there were no existing people, then the answer you gave is correct.

The integration adds the number of people born during a specific very small interval of time from time = 0 to10. That interval will see more people born as the time approaches 10.