Question

Could anyone explain me why change in Gibbs energy is 0 when the system is at equilibrium?

Answer 1

do you want a mathematical explanation or a conceptual one?

Answer 2

when equilibrium is attained there is no furthur free energy change. the reaction and its reverse are ongoing at equal rates. hence, there's no loss or gain \(\Delta\)G.
In other words, you have: \(\sf \color{}{RT + ln(K)= \Delta G = 0}\)