what is "one-to-one reduction from a function f to another function g"
The definition is composed of two parts. First, we want a reduction from $f$ to $g$. That's a way of computing $f$ given an oracle to $g$. This reduction is formed by two polytime functions $\alpha,\gamma$ which are used in the following way: $f(x) = \gamma(g(\alpha(x))$.

Second, we want the reduction to be one-to-one. That explains the if-and-only-if in (2). Moreover, for some reason we want an efficient one-to-one right-inverse $\beta$ of $\gamma$, satisfying $y = \gamma(\beta(y))$. The function $\beta$ can be used to answer the following question: what value of $g$ corresponds to a given value of $f$? (In contrast, $\gamma$ tells us what value of $f$ corresponds to a given value of $g$.) Why this is need will be apparent from the way the paper uses $\beta$.

The actual properties stated in the definition are slightly weaker (you want the function to be one-to-one only for values which actually occur in the formulas), but this is the gist of it.