perlop (Gory Details of Parsing Quoted Constructs): "The most important Perl parsing rule is the first one discussed below: when processing a quoted construct, Perl first finds the end of that construct, then interprets its contents."

m// is a quoted construct. So the "end" will be found before the contents are interpreted (including interpolated). Now if you say, "m/a/b/", the problem is that the end is found between 'a' and 'b'. But when you say, $c='/', and then m/a${c}b/, the end is found after 'b', and later the interpolation of $c takes place.