The Stone Age is the age when man used mostly stone - especially flint - for his tools. The early stages of human development were named by the early archaeologists of the 19th century after the dominant material used for tools and weapons: stone, bronze and finally iron. This works only B.C. and historical ages were named differently.
The namesake flint is quartz (SiO2), which is very hard and has sharp edges, especially when its broken. So it is usefull for many things: knifes, axes, scrapers and much more. Pieces of flint were glued to sticks to produce arrows, spears, or axes. The craft to produce tools from flint was developed during the Stone Age and as stone does not decompose, archaeolgic sites from the stone age are full of them, while biologic matter like the wood, the glue, leather, and baskets and ropes made of plant fibres are mostly destroyed. We know they existed, because under rare conditions some were preserved, for example in swamps or in permafrost.
After some research during the 19th century this age was divided into three periods which are defined by the abilities of man to produce more sophisticated tools. He learned to polish the rock and to drill holes into rock. Those periods were simply called old stone age, middle stone age and young stone age, but it sounds much better if you translate it into Latin.