Here, $P_S$ is a projective cover of a simple module $S$. This means that there is a surjection $\sigma\colon P_S\to S$ from projective $P_S$ onto $S$, which has superfluous kernel $K$. The fact that $P_S/K\cong S$ is simple implies that $K$ is a maximal submodule of $P_S$, and the fact that $K$ is superfluous then implies that every proper submodule of $P_S$ is contained in $K$. In particular, the only simple quotient of $P_S$ up to isomorphism is $P_S/K\cong S$.
In the underlined part of Lemma 2.2, $g\colon P_s\to S_r$ is a nonzero homomorphism onto a simple module. The kernel must be $K$ and we must have $S_r\cong P_S/K\cong S$.