I was reading these notes and on page 88 (of the paper version) it said:
however it wasn't clear to me why the function would be unique (or why it mattered). It seems entirely possible depending on the definition of $G$ that it might not be unique. Also, how does the explanation of making the explicit point by point what $F$ is explain anything? I think that is the key point I don't understand.
It should be pretty intuitively clear why $F$ is uniquely defined. If you want to know what $F(a,b)$ is for any $a$ and $b$, you simply apply the $F(a,b+1)$ equation repeatedly until $b=0$ and then you apply the $F(a,0)$ equation. This will unfold $F(a,b)$ out into a $b$-deep nested applications of $G$ to itself. Now, we'll still have occurrences of $F$ in this unfolded expression, but since they are always applied to smaller values of $b$ we can use strong induction to show that they too can be rewritten in terms of $G$ only. The end result is that $F(a,b)$ is defined as an expression solely in terms of $G$, $a$, and $b$. If you do the proof, you'll find that strong induction fits extremely nicely. This is no surprise as this is the computational analogue of strong induction (with parameters) itself.