I am unsure what exactly the "conclusion" of the theorem means, as in, is it the entire theorem without the assumption of normality of X? So I am assuming that we have to go the other way around as compared to the proof of Tietze's theorem, i.e, assume the function f on (closed?) subspace F has a continuous extension defined on all of X (both of who's values lie in [a,b]) and go backward to prove X is normal. Also, in particular it mentions $T_1$-space, why not a general topological space?
The entire proof of Tietze's Theorem is based on the assumption of normality of X, so I cannot understand how to go about proving the reverse to arrive at the condition of normality.