It depends entirely on how you define the function $x \mapsto \mathrm{e}^x$. For example, looking at other answers to this question:
- robjohn defines the exponential function in terms of the limit $\lim_{n\to\infty} \left( 1+\frac{x}{n} \right)^n$—from this definition, the intuition is that the derivative can pass through the limit, and then the power rule gives the desired result;
- S. Maths defines the exponential function via a power series—from this definition, it is again possible to pass the derivative through the limit (i.e. the limit can be brought into the sum) and then term-by-term differentiation gives the result;
- J.G. doesn't quite give a definition of the exponential function, but looks at the difference quotient directly and uses the "fact" that $a^0 = 1$.
Personally, I like to define the exponential function by the property that it is its own derivative. More precisely, the exponential function $x \mapsto \exp(x)$ is defined to be the unique solution to the initial value problem $$ \begin{cases} u' = u \\ u(0) = 1, \end{cases}$$ where $u$ is defined on the real numbers. Note that this initial value problem exactly encapsulates the idea of "constant growth"—the growth rate of a population is equal to the size of the population; each individual reproduces at a constant rate.
The "intuition" I have about this is that there should be some function which behaves this way with respect to differentiation—in a fair and just world, there should be a function which is its own derivative.
But then the question is, perhaps, why does this function look so much like exponential functions that are taught in high school? I was taught, at one point or another in my education, that if $a$ is a real number and $n$ is a natural number, then $$ a^n = \underbrace{a\cdot a\cdot \dotsb \cdot a}_{\text{$n$ times}}; $$ that is, exponentiation is repeated multiplication. What does $\mathrm{e}^x$ have to do with repeated multiplication!?
Typically, some of the first things that are done when the exponential-is-repeated-multiplication function is introduced is to show that it satisfies certain properties:
- $a^{m+n} = a^m \cdot a^n$,
- $(a^m)^n = a^{mn}$,
- $a^{-1} = 1/a$,
- $a^0 = 1$,
- for fixed $a > 0$, the function $n \mapsto a^n$ is increasing,
and so on. Note that some of these require some adjustments to the possible values of $a$, so the definition of the exponential-as-repeated-multiplication function is typically extended a little to make this all make sense. Then, after a bit more work, it can be shown that $\exp(x)$ has all of the same properties—this has already been done on this site, so I won't repeat the arguments here. This implies that there is some real number (call it $\mathrm{e}$) such that $$ \exp(x) = \mathrm{e}^x,$$ where $\exp$ (on the left) is the solution to the initial value problem, and $\mathrm{e}^x$ (on the right) is an appropriate extension of the exponentiation-as-repeated-multiplication function.
Note that all of the above is a bit hand-wavy (the question is about intuition, not rigor, after all), but it can all be made rigorous with a bit of work.
TL;DR: The exponential function is the solution to an initial value problem, and from this definition, the exponential function is its own derivative. A function which is its own derivative has all of the same properties as a particular exponential function (as defined in high school via repeated multiplication), and is therefore the same thing.