let f(ax, ay) = a^k f(x,y) be a homogeneous function in the plane x,y for a scalar a, then f can also be written as:
f(x,y) = f(y*x/y, y) = y^k f(x/y,1)
and the symmetry by a matrix transformation in the plane
x-> ax + by and y-> cx+dy then transforms the function f as:
f(ax+by, cx+dy) = (cx+dy)^k f((ax+by)/(cx+dy),1)
now introduce z=x/y and f(x/y,1)=F(z) then
F((az+b)/(cz+d)) = (cz+d)^k F(z)
Projective spaces are cool ;-)
just my 2 ct
I forgot to add to my post that, SL(2) is the symmetry group of the Riemann sphere. So since you often use this as your space in complex analysis, this symmetry transformation is everywhere. So that's one reason why you might expect to see modular forms in lots of places. (At least, that's what I understood, maybe a mathematician would tell you differently.)
Now make M to be a Mobius transformation, so that f((ax + b)/(cx + d)) = (cx + d)^k f(x), where the coefficients are in SL(2,Z) ie. integers with ad - bc = 1. Mobius transformations like this are common.
That's more or less it as far as I'm aware. There's some growth rate condition too, I don't think it's as important to an intuitive understanding as the transformation law.
Disclaimer; my training was in mathematical physics not mathematics. To me, what l wrote here was enough for me to feel like I understand what they are, at least to some basic level.