As mentioned in other answers, the starting point is the definition of the exponential you are using.
One possible definition is as the solution of $f'=f$ taking value $1$ at $0$ (or other equivalent phrasings if you don't want to involve differential equation explicitly) and then there is nothing to explain.
One more natural definition is that $\exp(x)=e^x$ where $e$ is some number (then you have to explain powers by a real number, which maybe tricky, but can be done with various level of details). This is the definition I recommend for people that will have to compute with exponential and logarithms, because it embeds the morphism property $e^{x+y}=e^x e^y$.
The number $e$ is somewhat difficult to justify a priori, so what you can aim for is explaining why an arbitrary exponential function $x\mapsto a^x$ is always proportional to its derivative (and then it is easily accepted that for some choice of $a$ the proportionality coefficient is $1$, and that makes an excellent definition of $e$).
If you aimed at a computational approach, you would write
$$ \frac{a^{x+h}-a^x}{h} = a^x\frac{a^h-1}{h}$$
and then one sees immediately that if $(a^h-1)/h$ has a limit when $h\to 0$ (a small leap of faith), then $x\mapsto a^x$ is indeed proportional to its derivative.
But you ask for a geometric approach, so let us make this geometric. Draw a plausible graph for an exponential, and translate geometrically the definition of an exponential: if we move on the graph by adding any value $h$ to the $x$-axis, we obtain a fixed relative increase, i.e. the $y$-coordinate is multiplied by an amount that only depends on $h$. This exactly tells you that the slope of a chord from $(x,a^x)$ to $(x+h,a^{x+h})$ depends on the point, but the ratio of the slope to the $y$-coordinate of the leftmost point only depends on $h$. Passing to the limit, you get the desired property.
Note that if you where to define the exponential with the differential equation, this explanation could help understand why we end up with a function that takes a fixed value to the power the argument.
In any case, I think the chosen explanation should be an opportunity to manipulate the morphism property of the exponential, because it is tremendously important and often mistreated by students.