I think the reference is to the oxygen that was in already in the water, or introduced by mixing and pouring. Leaving it in an open bucket would give it an unlimited supply of oxygen and cause it to go bad sooner.

I mix it in a plastic milk jug (using half the water). When it dissolves, I add the rest of the water to make the correct solution, mixing more as I go, and cap it air-tight. Then I let it sit at least a day before using.

Once I first use it, I divide the remaining developer into 1-litre pop bottles, filled to the top, and capped air tight.