Question

why does the electron gain enthalpy increases across the period

Answer 1

as you go across the period~ the numbers of protons in each element is increasing~
increasing the numbers of protons increase the strength to hold on to the electrons~
(protons = +ve charge ; electrons = -ve charge thus attracted to each other by the electrostatic force)

since the strength between the protons and electrons increases, the energy needed to pull an electron away from the nucleus increases, so the enthalpy increases.

hope that helps ;)