The quick story is that nobody knows exactly, because potter's wheels have existed since prehistory.

However, prevailing thought is that the modern wheel came to be through several successive designs:

1. A potter walking around a fixed pot, building it with coils
2. A potter turning a pot on a crudely revolving base
3. A potter spinning a pot on a wheel, supported by spindles, bearings, or similar methods

This all happened before the year 2000 BCE-ish, originally in only one location in the near east, and spread from there.

Bernard Leach, the famed English potter, talks about this in his A Potter's Book. Here's a quote from the relevant section, expanding on the above:

In Japan I watched one of the last craftsmen able to build up a large pot without using a wheel. The potter walked backward round a jar, 3 feet high by 3 feet in diameter, with his right forearm loaded with a rope of clay 18 inches long and 2 inches thick, coiling, squeezing and smearing it onto the growing wall, which was supported on the outer side by the palm of his left hand. The movement was regular and rhythmic, and even before he proceeded to heat the fresh piece of work with a bat, replacing his left hand with a shaped wooden support, the surface was surprisingly smooth and even. Later, when he took wet cloths in each hand and smoothed the six inches which he had added to the height of the pot, it was difficult to believe that the pot had not been thrown on a wheel. Amongst some primitive peoples a crude revolving base is used to overcome the necessity for walking round the pot, and it seems evident that this is the origin of the potter's wheel.

The wheel upon which a potter moulds his hollow forms is one of the most peculiar and intimate devices which human beings have invented. There is nothing quite like throwing in any other craft. Wood, metal, fibre and glass are none of them so responsive to the touch as clay.

The potter's wheel is known to have existed in Egypt from the commencement of recorded history, and in China and Europe it dates back to at least 2000 BC. The consensus of opinion is that it originated somewhere in the Near East, and that the knowledge of it spread from one centre. The drawings illustrate a few of the many types of potter's wheels. In principle they all consist of a wheel-head, or disk, which revolves with considerable momentum and smooth control of speed and is driven by hand or foot or by gear. The momentum is obtained from either a heavy wheelhead or from a flywheel. In some cases the spindle is attached to the wheel and revolves in a socket at the base, in others the socket is in the centre of a hollow shaft attached to the under part of the wheel head and the spindle is fixed in the ground. The latter type is common in the East and the former in the West. The primitive kick wheel is turned by the direct action of the bare feet on the flywheel, but the more evolved type has a crank in the iron shaft with a kick-bar attachment. Before the coming of mechanical power, wheels were also driven by a belt from a large separate wheel hand-turned by an assistant.

Source: Leach, Bernard. A Potter's Book. Faber and Faber, 1940