The formula :
$\vdash_T G_T \leftrightarrow \lnot \exists y$ Prf$(\ulcorner G_T \urcorner, y)$
it is provable in $T$ the equivalence between the sentence $G_T$ and the sentence $\lnot \exists y$ Prf$(\ulcorner G_T \urcorner, y)$ [see also this post].
Assume that we know what is a Formal system like First-order theory of arithmetic based on First-order logic.
If the system $T$ has "certain arithmetical capabilities" (to be specified) we can use them to build up formulae related to the syntactical facts of the theory itself; see Arithmetization of syntax.
One of this formulae is the "proof predicate" Prf; we can define it in $T$ such that Prf$(\ulcorner G_T \urcorner, y)$ holds when $y$ is the "code number" (called Gödel's numeber) of a proof in $T$ of the formula of $T$ whose "code number" is $\ulcorner G_T \urcorner$.
For the details, you need an exposition of Gödel's Incompleteness Theorems; I assume that you can find it in your textbook.
It is usually included in all "major" textbooks, like :
Elliott Mendelson, Introduction to Mathematical Logic (4th ed - 1997)
Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001).
In addition, yu can see this fully-dedicated book : Peter Smith, An Introduction to Gödel's Theorems, Cambridge UP (2nd ed - 2013).