The result $$\zeta(2) = \frac{\pi^2}{6},$$ tends to amaze young students because of its beauty.
However, although in literature there are many proofs of this result, I find that none is suitable for someone who didn't take some advanced classes in calculus. Moreover, as far as I know, they do not offer an intuitive explanation of why this result should be true.
So my question is:
What is the key intuition -- that is, the picture -- behind the result? Are there any visual of anyway intuitive proofs (*) of the statement that can be used to convey a better (or, at least, different) understanding of it (to people who haven't taken advanced calculus yet)?
(*) Note that full rigour is not compulsory for the scope of this question (it is very appreciated by me personally, though).