As the name says, reproducing kernel Hilbert spaces is a Hilbert space, so some knowledge of Hilbert space/functional analysis comes in handy ... But you might as well start with RKHS, and then see what you do not understand, and what you need to read to cover that.
The usual example of Hilbert spaces, $L_2$, have the problem that the members are not functions, but equivalence classes of functions that coincide except on a set of (Lebesgue) measure zero. That way, they always give the same results when integrated ... and that is what $L_2$ spaces can be used for. Members of $L_2$ spaces cannot really be evaluated since you can change the value at one point without changing the value of the integral.
So in applications where you really want functions that you can evaluate at individual points (like in approximation theory, regression, ...) RKHS come in handy, because the defining property is equivalent to the requirement that the evaluation functional
E_x(f) = f(x)
is continuous in $f$ for each $x$. So you can evaluate the member functions, and replacing $f$ with some other function, say $f+\epsilon$ (in some sense ...) will only change the value a little bit. That is the intuition you asked for.