Mathematically, the logistic map is written as follows
$$x_{n+1}=rx_{n}(1-x_{n})$$
Here, $x_n\in[0,1]$ represents the ratio of the existing population at each step ($n$) to the maximum possible population.
Now, in nature, population growth is often limited by factors such as available resources, competition, and space as the population approaches the carrying (maximum) capacity. This limitation is typically modelled as a reduction in the potential for growth as the population gets closer to such capacity.
To represent this limiting effect in the logistic map, the term $1−x_n$ is introduced.
This choice seems somewhat arbitrary though. Why not a negative exponential, for example? Isn't there a rigorous derivation from first principles?
Depending on what you are interested in, an exact knowledge of the map is not necessary. For example, if you are interested in the period doubling route to chaos, you only need that the map is unimodal with a non vanishing second derivative at the maximum. Even its quantitative features (Feigenbaum constants, fractal dimension etc.) do not depend on the map away from the maximum. This is what is referred to as universality, and justifies the sloppy justification of the map.
Hope this helps.