I guess one answer to "meaning" is that, for example, weight 0 modular forms of level 1 (i.e. the definition that you have there, with k=0) correspond to functions when you mod out the upper half plane by the action of $SL_2(\mathbb{Z})$ (the resulting space, if you compactify it by including cusps, is the modular curve $X_0(1)$ - which is isomorphic to $\mathbb{P}^1$ - so it's no surprise that functions are just constant (i.e. weight 0 modular forms are just constants).

Modular forms of weight 2k corresopnd to k-fold holomorphic forms on $X_0(1)$ (so for example there are no weight 2 modu…

@Daminark It's possible that I may have made some mistake in what I said though (unfortunately I don't know this off the top of my head all that well..). But you can look at section 2 in https://www.math.wisc.edu/~hast/Notes/Math847-modular-forms-notes.pdf , or more generally "modular forms" by Miyake ( in particular chapter 2). The modular interpretation can be found in Diamond and Shurman's book.

But this story doesn't quite explain (at least I don't see it) all the other miraculous stuff that modular forms somehow prove, e.g. ones that @MatheinBoulomenos is talking about.

Hello!!

A DIN A4 sheet is divided into thirds. A rectangle is the root of the tree, the other two rectangles are each divided into thirds again. Two rectangles form the branches - one to the left, one to the right - the others are again divided into thirds and so on.

How can we calculate the area of that tree?

Do we maybe do the following?
Let x be the area of the paper, then we divide this into 3 parts, so into three parts ith area x/3.
Then each of the two parts is again divided into three parts, i.e. now we have x/3 + 2*(x/3)/3.