How Separation of Variable method eleminate all the Canonical Variables in Hamilton Jacobi Equation?

Question

In the derivation of Hamilton Jacobi Equation, it was written that with this Separation of Variable method (highlighted in red), Equation 2 can be splitted up and eventually all the variables $Q_i$ will be replaced by constants $\alpha_i$. I don't get the procedure. How does "function of $\left(q_1\right)=$ function of $\left(q_2, q_3 \ldots \ldots \ldots q_f\right)$ not involving $q_1$" come from? and should not $q_i$ instead of $Q_i$ be replaced by constants $\alpha_i$?