Three thousand years ago a bunch of Chinese silkworm farmers got fed up with their job. Instead of carrying out the tedious task of harvesting hundreds of silkworm cocoons for their silk, the farmers wondered if there wasn't an easier way they could make the stuff artificially. There was, but the techniques wouldn't be available for another few millennia.
At about the same time, over in ancient Greece, the architect/inventor Daedalus was also inspired by nature. According to myth, Daedalus was languishing in the labyrinth of King Minos with his impetuous son. Inspired by the views from the labyrinth of the birds flying overhead, he made artificial wings from feathers held together by wax, and flew from his prison to freedom (his son Icarus wasn't so lucky, but that's not the part of the story we're interested in today).
In a better documented, if more mundane example, the Swiss engineer Georges de Mestral invented Velcro in the 1950s when he realized how the hooks of burrs clung to his dog's fur.