If we can simulate $NP$-machines why isn't it trivial that $NP=coNP$?
When we say "Simulate $M_A$ on the query string $s$", we simply mean execute $M_A(s)$. Note that you construct a non-deterministic machine for $L$, thus you are able to simulate other non-deterministic machines. If both $M_A(s)$ and $M_{\overline{A}}(s)$ output $0$, then stop and reject $x$ (the initial input). Only when one of the machines outputs $1$, you can be sure about whether or not $s\in A$, and can continue simulating the oracle machine for $L$.

Note that this newly constructed non-deterministic polynomial time machine, $M(x)$, may accept its input $x$ only in computations where for each query $s$ raised during the computation, either $M_A(s)$ or $M_{\overline{A}}(s)$ output 1. Using this observation, you can simply show that $M$ indeed decides $L$.