It comes from a 1970's british TV commercial for Lemon Squeezy detergent. They were with a little girl who points out dirty greasy dishes to an adult (mom or relative) and then this adult produces Lemon Squeezy and they clean the dishes quickly. At the end of the commercial the girl says "Easy Peasy Lemon Squeezy".
Today it is a silly way to state something was or will be very easy.
I will be in an out, easy peasy.
Can you open this jar of pickles? Sure thing easy peasy.