What is explanation of $\text{Sat}([φ],s)$ in the definition of Rayo’s number? The definition of $\text{Sat}([φ],s)$ can be found here.  
All I want is an explanation of what each line in this definition means and how $\text{Sat}([φ],s)$ works. The only relevant thing that I have found is the following comment ( source ):  

He is referring to truth in the set-theoretic universe (not the natural numbers), and the class R is a truth function, for the clauses in the definition assert that R obeys the Tarskian recursion, and the final clause asserts that phi is true of s, according to the truth predicate R.

 A: That is the satisfaction predicate. It takes a formula and an assignment to the variables of the formula, and tells you whether or not the statement is true or false. 
Rayo's number is really just a complicated way of saying that it is the smallest number which larger than all those numbers definable in the set theoretic universe with a formula of at most googol symbols.
The catch, of course, is that almost always we talk about the natural numbers as their own entity, so naming large numbers is taken by talking about some function like $\rm TREE$, or Busy Beaver, or so on. But the mathematical universe is so much larger. The set theoretic universe gives us tools much larger than just the natural numbers.
It also opens the door to the ills of independence, at least when playing this game of naming the largest number (e.g. $n$ is the least integer such that $2^{\aleph_n}>\aleph_{n+1}$, or $0$ otherwise). Which is why we need to refer to a fixed universe, and to the satisfaction relation of that universe.
