How is there not a single quote from Anush in the article? She is one of the coolest mathematicians I know, which makes this article a major lost opportunity.

Then that means there are at most ~ 2^{s(n)^2} circuits of size s(n). For s(n) small, like a polynomial, this is way way smaller than the number of total Boolean functions, 2^2^n

Take any circuit C of size s(n). How do we encode it as a binary string? For every gate, it has a Type (and, or, not, input, output) which takes 3 bits to describe, and then you can have a Yes/No bit saying if gate i feeds into gate j for every pair i, j < s(n). This is a string of length ~ 3s(n)^2

To rephrase, you are asking, what is an intuitive way to count the number of circuits of a specific size, s(n)? Size here we usually count as the number of wires appearing in a circuit (though number of gates is fine too).
The easiest way to get a growth rate upper bound is an encoding argument.

You could do a dovetailing over tuples (l, m, b) where l is the l-th Turing machine, m is the exponent of the running time n^m you simulate the machine for, and b a 0/1 value for giving as input phi or not phi. Simulate and see if a valid satisfying assignment is output.

Can't wait to read it!