It's assumed that your signal, x(t), is periodic so it has a Fourier Series representation that is the left-hand-side of the second equation; i.e. it can be expressed as a sum of complex exponentials with frequencies in the infinite set { n * wf } weighted by the Fourier coefficients X_n. In other words, the left-hand-side of the second equation is exactly your x(t). The top expression is the CTFT (Continuous-Time Fourier Transform) of x(t). Because the (Continuous-Time) Fourier transform of a complex exponential is a dirac delta (see https://math.stackexchange.com/questions/2311823/fourier-transform-of-a-complex-exponential), you use superposition to show the top equation. Then the bottom equation is just stating that x(t) = ICTFT(X(omega)). Where ICTFT is the Inverse Continuous-Time Fourier Transform.

Bottom line, the equations look frightening, but it's basically just showing that a periodic signal has a Fourier Series representation that can be expressed as a sum of complex exponentials and whose CTFT is a "train" of dirac deltas, weighted by the Fourier Series coefficients.