The first step was the development of a way to draw metal strings, instead of twisting strips and hammering them out. This happened around 1300, and the result was the creation of a number of new stringed instruments. First of them them that whacked the strings with sticks were hammered dulcimers: some of these were simple string drums ( much like the modern tambour de Bearn) and some were more complex instruments that could play melodies, hexachords. But fairly soon, at least by mid 15th c., there were harpischords and clavichords, keyboard instruments that plucked or hammered the strings. The first clavichords seem to have had little more than keys as levers: push down the key, the back end pivots up and hits a string. Even though this basic instrument lasted for hundreds of years, especially as a practice instrument, it had major limitations for controlling sustain, dynamics, etc. and clavichords , including a kind called a fretted clavichord, were pretty quiet things... Here's an example of one being played.

An Italian named Cristofori came up with the first effective piano action circa 1700, and that is usually considered the first piano (If you follow the links, here, one will how you a diagram of his action).

But it was not until the later 1700's that the piano really took off, with the development of various different actions by makers like Broadwood in England, Pleyel in France, going into the mid 1800's and beyond. So what you see in that animation of a modern grand action is the end product of quite a lot of experimentation for quite a long time. It should be noted, too, that this is only grand action, which depends on gravity to return the hammer. Upright pianos have been an even more recent development , and the upright piano action has perhaps even more complexity in order to have an adequately quick return of the hammers.