For a laymen to military history like me, chariots don't seem nearly as good as horse archers. Chariots needed to be crafed, very expensively, often needed two people, a driver and an archer (or sometimes three in Assyria, a driver, shieldbearer, and the archer) to function. Given their size, you can fit relatively few archers on any given size of an area (even if moving)
In comparison, a horse archer takes half the space, so you can have more of them at mentioned given area, doesn't need costly and hard ot maintain contraptions, and it's only one guy you have to pay/force in to service.
What am I missing?
There were a few reasons for a transition. It would take less training with a chariot. The archer and the driver can focus on what they're doing. The earliest horse archers were a pair of horses with one rider holding the reigns of the archer's horse to make it work.
Although not the only innovation, a major change was the development of stirrups. Without stirrups it is much more difficult to ride, let alone do anything while riding. The level of skill involved meant it was only possible realistically for people who lived in regions where horses were indispensable and they grew up riding.
Once stirrups were common, it became much easier for someone to control a horse and have the range of motion to make archery much easier. It would still take a great deal of skill to do it effectively however. (If you're the type who's done much riding, once you're actually on a moving horse the difference between stirrups and not is night and day.)