How would you explain why "e" is important? (And when it applies?)

There is also a difference between an interest in understanding

• the mathematical constant "$e$", which is an irrational, transcendental number, but real number, just as is $\pi$, but a number, nonetheless, which happens to be the value of your limit, and happens to be the value of the infinite sum $$e = \displaystyle \sum_{n=0}^\infty \frac{1}{n!}$$And more: $$e^x = \sum_{n=0}^\infty{x^n \over n!} = 1 + x + {x^2\over2!}+{x^3\over3!}+{x^4\over4!}+\cdots$$ vs.

• exponential functions (base $e$: (e.g., $f(x) = e^x$, or $e^{i\theta} = \cos \theta + i\sin \theta$), functions to which you seem to refer when you speak of its use in representing exponential growth, and other applications which crop up everywhere, so it seems. Certainly, functions involving $e$ tell us something about $e$, but they tell us so much more than that.

That is, the importance of $e$ itself isn't so much for its significance as a particular real number, but its significance as a base for exponential functions, and in terms of the ways that functions involving $e$ as a base appear in surprising ways, are powerful, and have applications in many domains.

So it's hard to know exactly what you are interested in: the number $e$, or the many functions involving $e$.

Not an easy task (the teenage part).

Any system where the rate of change of a quantity is proportional to the amount of the quantity has solutions involving exponentials. This is true for continuous as well as discrete systems. Chemical reactions, electronic circuit behavior, physical systems, etc, are often well approximated by such systems.

If you differentiate an exponential function, you get itself times a constant: $$ \frac{d}{dx} 2^x = \left(\text{constant}\cdot 2^x\right). $$ In other words, the function grows at a rate proportional to its present size.

Only if the base of the exponential function is $e$ is the "constant" equal to $1$, so that you get $$ \frac{d}{dx} e^x = 1\cdot e^x. $$ In other words, the function grows at a rate equal to its present size.

It's the same as the reason why radians are used in calculus. You have $$ \frac{d}{dx} \sin x = (\text{constant}\cdot\cos x). $$ If you use degrees, the "constant" is $\pi/180$. If you use radians, the "constant" is $1$, but only if you use radians.

There's more to the story than that. For example, how does the Poisson distribution arise as the limit of binomial distributions? But the above should show you in what sense $e$ is "natural", and in what sense radians are "natural".