The "double standard" requirement might seem at first like doing two opposite things.
To understand it, let's explain it that way:
You have a pot without a drainage hole. You water it. The water table reaches the surface of the water. Soil components (peat, organic matter, etc...) begin to rot. The decomposition process steals oxygen from the soil and anaerobic processes begin to work. The roots have no chlorophyll and don't photosynthesize. They suffer from oxygen starvation and die.
On the other hand, hydroponic systems seemingly don't have this problem. Why? Although the roots may be completely submerged in water, water is being actively oxygenated. In addition, hydroponic systems use little to no freely-floating organic matter which decomposes and consumes oxygen. The roots have both water and oxygen. Some hydroponics growers have taken that one step further and created aeroponics: The roots are dangling inside a container and are periodically misted with water spray to keep them wet while at the same time get plenty of air.
Now we can apply this approach to potting mixes: The mix has two physical characteristics: water retention and grain size.
Water retention may be a physical property of the mix, depending on what it is made of. It normally refers to water absorption. Water-absorbing grains, aiding retention, may store the water within the grains for later consumption by the plants, leaving the space between them aerated.
Grain size may affect aeration as well: Water holds on to the grains by surface tension. If the grains are too small, the space between the grains will be filled with water and hardly any air. If the grains are large enough, there will be lots of space for air while water will be confined to the grains which release it gradually, so the roots get both air and water. Excess water cannot hold by surface tension and drains away. This is the ideal case. Normally, an additional component such as Perlite aids in aeration because it retains less water than peat or coir, for instance.
A word of caution: To keep those two parameters at satisfactory levels, it is advised to use coarse-grained medium. Degraded (old / crushed) soil normally loses those properties. It becomes too water-retentive (within the space between the grains) and does not allow enough air. Normally, manufacturers mix two or more materials to reach a "middle-point" of these parameters. Professional growers may also make special potting soil suited for unique plants, but this is another story.