Suppose that I went to Tasmania a few years before the "Tazie Tiger" (thylacine) became extinct. I sample say, $100$ thylacines and make some biometric measurements. To make the discussion concrete, let's make the data the skull widths at the widest point on the head.
From this data I can calculate the sample mean, $\bar x$, and the standard deviation, sigma.
I get confused by the use of $n-1$ in the denominator of $\sigma$. So, I go away, look at Wikipedia, find out about Bessel's correction, slowly work my way through the maths and eventually learn to accept that, since $\bar x$ is calculated from the data, I only have $99$ independent comparisons of $x-\bar x$.
This makes a kind of sense. If I only have $1$ observation, there are no other data to compare it with. If I have $2$ observations, I have exactly $1$ estimate of the spread of the data, which is the spread of the data, and so on. I even understand that the sum of the absolute deviations must equal zero as the mean has been calculated from the data.
Great! I go away, happy with my calculations.
A week later, someone tells me that I was so thorough with my survey that my "sample" was actually a census. I have taken measurements of absolutely every single living thylacine in the world. Suddenly, my sample average, $\bar x$ is actually the true population average $\mu$. Clearly, I now have to use the population standard deviation.
I go back to my calculation. The number of data is still $100$. The population average has still been calculated from the data. The sum of the absolute deviations still adds up to zero. I still have $n-1$ independent estimates of $x-\bar x$.
The data are identical. The algebra is identical. So when I calculate the standard deviation, why do I now divide by $n$, rather than by $n-1$? Where did the extra degree of freedom come from?