This video on the English Heritage channel demonstrates how prehistoric pottery was created in Neolithic Europe.  It is shown how the clay is formulated, then formed into pots, and then baked in fires.

[Note: I think the inexplicable delay of the adoption of the creation of pottery is a big mystery of prehistoric humanity.  Hominids have been harnessing fire for well over a million years, with their entire lives being centered around the use of it.  What stopped them from setting clay objects into the fires during that time?  Wouldn’t at least some of such objects accumulate in the archeological record?  Also, people abruptly discovered how to create metal objects by smelting gold, copper, and then bronze relatively quickly after pottery was adopted, which is significantly more complicated that simply baking mud to harden it.]