Imagine you have a function which transforms nicely under scaling, ie. f(Mx) = M^k f(x) for some b. Then if you have systems where you use x -> Mx regularly, you'll find these functions turn up because they're the ones that respect the symmetry.

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.