History of Baskets
Chimpanzees don't weave baskets, but people probably learned how to make baskets very early on, soon after they became people, before the first people left Africa. The earliest known baskets were made about 27,000 BC in eastern Europe, but people probably made baskets much earlier - they just don't last very well, because they rot.
Aztec women with food in baskets, ca. 1500 AD
Baskets are cheap and easy to make, and useful for gathering food, carrying things, and storing things, so everybody wove baskets, all over the world. People used baskets as backpacks, as baby carriers, as cupboards, as plates and cups, as cradles, as birdcages, as measuring cups and as sandals, and to catch fish. You can make baskets waterproof to hold water, and you can boil water in them by dropping in hot stones from the fire. You can weave a light wagon out of basketry.
People also used related techniques to weave straw mats to sit on and sleep on, and straw rugs for their floors. They wove straw hats to keep the sun off their heads, and they knotted nets to catch fish and small animals and birds. Soon this kind of weaving led to finer weaving which made linen and hemp clothing and blankets.
But even after the invention of cloth, people kept right on using woven baskets and straw or reed mats for a lot of their stuff. Plenty of free material - reeds and tall grasses and straw - was available all over the planet, and anybody could learn to make baskets and mats, even if specialists made better ones. Baskets and mats didn't take long to make, they were useful in thousands of different ways, and they could be made to be beautiful, too.