The s-m-n theorem
Suppose that $f(x,y)$ is a computable function (not necessarily total). Then for each fixed value $a$ of $x$, $f$ gives rise to a unary computable function $g_a$, where $$g_a(y)\cong f(a,y)$$ Since $g_a$ is computable, it has an index $e$, say, so that $$f(a,y)\cong\phi_e(y).$$
This is the definition, the issue is, I still don't understand the basics. Here is my interpretation. If we have a computable function $f(x,y)$, we can assign all the values of $f$ in unary form, with function $g_a(y)$, then I don't understand how we get that little "$e$" and the $\phi$ function. What does that little $e$ actually mean? In addition, why do we need this function form?