Suppose the original standard form of an optimization problem is:
$$\min~~f_0(\mathbf{x})\\ \text{s.t. } ~f_i(\mathbf{x}) \leq 0, i=1,...,m \\~~~~~~~ h_j(\mathbf{x}) = 0, j=1,...,p $$
in which $f_i$ for $i=1,...,m$ are inequality constraint functions and $h_j$ for $j=1,...,p$ are equality constraint functions. For a convex optimization problem, $f_0(\mathbf{x})$ and $f_i$ for $i=1,...,m$ are convex functions and $h_j$ for $j=1,...,p$ are affine functions.
Now, for the equivalent epigraph representation of the original problem in standard form, we use the corresponding constraint in inequality form, we have:
$$\min ~~t \\ \text{s.t.} ~~ f_0(\mathbf{x}) - t\leq 0 \\ \qquad f_i(\mathbf{x}) \leq 0, i=1,...,m \\~~~~~~~ h_j(\mathbf{x}) = 0, j=1,...,p.$$
Assume the original problem is a convex optimization problem. To provide the original problem in epigraph standard representation, but preserving the problem to be in convex form, it needs to add an inequality constraint function $f_0(\mathbf{x}) - t$ which is convex in $(\mathbf{x},t)$; this inequality corresponds to the $\mathbf{epi} ~f_0$, here, is a convex set. So, the problem in the equivalent epigraph representation is still in a standard convex optimization problem form. Furthermore, for straightforward and meaningful analysis of a problem, also designing an efficient algorithm, different equivalent representation of a problem can be used.
Note: $f$ is convex if and only if $\textbf{epi} ~f$ is convex set.