I think I can explain in a bit of detail why you reject the first egg when you start breeding (sorry if it has been explained before).
The basic matter is because from what I have found, assuming you save before you put your Pokémon into the day care, take the first egg, hatch it, then reset, & do it again, you will find (assuming the same mother) that the nature, the gender & the ability of the child will be fixed regardless of the male, & the IV's will be different, but only with different males. That is, if I breed the same Male Chesnaught with the same Female Diggersby for example multiple times only saving before I put the two in the day care, I will find that the IV's of the Bunnelby, as well as the already pre-determined ability, gender, & nature will be the exact same every time. And this is assuming both Pokémon are not holding items. Like for example, I have been getting the same Bunnelby in recent tests, that is Female for the gender, Quirky for the nature, Pickup for the ability, & "Capable of taking hits" for the characteristic, with stats of 11 HP/5 Atk/6 Def/5 SpA/6 SpD/6 Spe. This basically means the characteristics of the Pokémon are fully predetermined even before you put your two parents in, assuming the same parents each time.
By saving before rejecting the first egg then taking the second, you ensure that the next egg you take will always be random in nature, gender, ability, & IV's assuming no items. I have tested this multiple times with the same two parents as before, & have gotten wildly different Bunnelby each time. This is why you always reject the first egg, then take the next egg for the sake of soft resetting.
tl;dr, Fixed properties of the first egg means it is better to reject the first egg.