What S(m) = T(2^m) means is that S, T are two different functions which produce the same result while taking inputs as m and 2^m respectively.
Function S can be considered as an operator with two internal steps (otherwise, composition of functions):
S' operator: Input:m, Output:2^m
T operator(original function): Input:output of first part, Output: As defined originally.
Therefore then transitions are:
m->2^m->T(2^m) = S(m)
(m/2)->(2^(m/2))->T(2^(m/2)) = S(m/2)