Can someone please explain what "properly aged water" means? How long should it be aged for, what types of setups do people use for storing and aging water?
"Properly aged water" depends on what it's going to be used for. If it's going to be used for most fish, then to properly "age" the water, you simply add tap water conditioner and make sure that the temperature of the "aged" water is about the same as the temperature of the aquarium.
For fish and invertebrates that are highly sensitive to heavy metals and dissolved organic compounds, you can:
1) add a fast growing plant, and let it sit in the sun for a day or two, or
2) add a surface skimmer, add lots of plants and wait 3 to 4 days, or
3) add tap water conditioner that also neutralizes heavy metals (e.g. Aquasafe) and increase the frequency of water change).
Only fish (that are widely available) that you (may) need do this are elephant nose and baby whale.
And with all the water conditioners available (dechlorinators etc.) is it really necessary?
No. I've never had an elephant nose or a baby whale, but I've had rams and discus with only tap water conditioner (with frequent water changes). They did just fine.