I know that it is something super easy but I just cannot get it.
Chain rule: $\frac{dy}{dx}=\frac{dy}{dt} \cdot \frac{dt}{dx}$.
I believe that I just don't understand the notation in the problem. What does $\frac{d}{dx}$ stand for? It is a derivative of what? Any help would be appreciated.
$\frac{d}{dx}$ is just "the derivative (with respect to $x$)". If you "multiply" it from the right with a function, that represents taking the derivative of that function: $$ \frac{d}{dx}\cdot f=\frac{df}{dx} $$ One pragmatic view of this is that it lets us argue about different ways of taking derivatives of the same function without having to actually mention that function. But letting the operator $\frac{d}{dx}$ go beyond this pragmatic view and actually becoming a fully-fledged algebraic entity on its own has turned out to be useful, and it's a popular thing to do.
If you're uncomfortable working with it as-is, you can multiply from the right by an arbitrary (differentiable) function $f$, work with it the way you're used to, then factor out the $f$ again afterwards: $$ \left(\frac{d}{dx}\right)\cdot f=\frac{df}{dx}=e^{-t}\cdot e^t\frac{df}{dx}\\ =e^{-t}\frac{dx}{dt}\cdot\frac{df}{dx}=e^{-t}\frac{df}{dt}\\ =\left(e^{-t}\frac{d}{dt}\right)\cdot f $$ Since the first term and the last term are equal for any differentiable function $f$, the terms in the parentheses must be equal as well.