Why is this flow a max flow?
You've left out part of the statement.  It should be "If there's no path between the source and the sink with unused capacity the flow is a max flow."  If you look at your graph you'll see that there is no path with unused capacity all the way from $s$ to $t$.  The $s$ to $a$ link has spare capacity but $a$'s lone outbond link is saturated.  The $s$ to $c$ link is saturated.