A very clear explanation of uniformization by Schottky groups can be found in Ch. X of L. Ford's book, Automorphic functions, Mcgraw Hill, 1929.
The proof is not constructive. Riemann surface is a sphere with $g$ handles. Cut every handle, and you obtain a topological sphere with $2g$ holes. By a theorem of Koebe this is conformally equivalent to the Riemann sphere with $2g$ round holes. All known proofs of the Koebe theorem are highly non-constructive. Even in the simplest cases, it is a challenge to compute the generators of the group from an explicitly given Riemann surface. Ford's exposition has an advantage that it is very geometric and intuitive.
In the cite you give there is one incorrect sentence: the Riemann surface is not $C/\Gamma$. It is $\Omega/\Gamma$, where $\Omega$ is the discontinuity set of the Schottky group.