More formally, the theorem says that if $f\colon \mathbb{N} \to \mathbb{N}$ is a total computable function (which we view as a method that constructs a program $f(e)$ from a program $e$) there is some program $e_0$ such that the computable function with program $e_0$ is the same function as the function with program $f(e_0)$.
A trivial application: if $f(e)$ is a program that simply outputs $e$ and stops, the program $e_0$ outputs its own source code.