Time-shift confusion
Perhaps the easiest way to understand it is by seeing what happens as we plot graphs of an arbitrary sequence $x[n]$:



As it goes through the system and take one every second sample (i.e. $x[0]$, $x[2]$, $x[4]$, $x[6]$, $x[8]$ and $x[10]$ in the graph above), you get $y[n] = x[nM]$ which can be depicted as: 



Now when you shift the input you get $x'[n] = x[n-1]$:



If we again take one out of every second sample we would then get $y'[n] = x'[nM]$:



We then note that this corresponds to the sequence $x[1]$, $x[3]$, $x[5]$, $x[7]$ and $x[9]$ from the original graph. For this simple case where $M=2$ you can see that the result is
$$
\begin{align*}
  y'[n] &amp;= x'[nM] \\
        &amp;= x[2n-1] \\ 
        &amp; =x[nM-1] &amp;\mbox{given $M=2$}
\end{align*}
$$

A similar process could be followed to arrive at the general result whereby the output of that system given an input of $x[n-1]$ would be $x[nM-1]$.