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Because having just air means the air can move and readily transport
heat from the warm wall to the cold wall. With insulation filling
the cavity, there is still a lot of air there, but it's being held in
place by the insulation, so it's far more difficult for an air
molecule to go from one wall to the other.

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Exactly. There is an optimum for every insulation material, be it
fiberglass, cellular foam, cellulose, etc. One big space is not so good.
divide it into four and it is slightly better, divide again into four
hundred it is better, but divide it into 40,000 spaces it is great, but then
if you divide that into 80,000 spaces the solid material between the air
cells is transmitting the heat again. That is why concrete walls are so
poor compared to even thin layers of insulation.

Heat is always trying to move to the cooler places. With most materials it
moves through liquids faster than solids where it moves faster than air.