That’s some high IQ usage of a meme. Lemme see if I’m getting this right:

• the total area of the image ( = RHS of the equation) is 1
• you divide the image into 4 parts so that the area of 1 part is is 1/4 ( = 1/2^(2*1)^). You take the first three quarters and leave the fourth quarter for recursion (I’ll call it x~1~). That gives you 3(1/4) + x~1~ = 1
• now you take x~1~ and do the same with it. This time, the area of each sub-quarter is 1/16 ( = 1/2^(2*2)^). Three such sub-quarters and a leftover x~2~ gives you 3(1/16) + x~2~ = x~1~. Put this back into the first equation to get 3(1/4 + 1/16) + x~2~ = 1.
• repeat until infinity; each time the area of the resulting tile is 1/4 of the previous tile (which is the 2n in the exponent part)