Why, after 2,000 years of artless millennia did people begin to draw cave art?

Because it was a new way of telling and recording stories. More than likely what we know of as the first art wasn't really the first art. For hundreds of years before the first marking were made on cave walls, people probably made markings in dirt and sand as a part of their oral tradition. Then one day, somebody came up with a more permanent way of making art. The rest is literally history.

assuming that it won't offend you to refer to evolution, I will. People evolved from primates. So initially human beings didn't have the capacity to create something like art. With the idea of evolution, they started out as animals with their brains functioning for survival (food water and shelter) and necessities alone.

Now we know humans these days are very different.

At some point between these two steps, humans began to feel a need to create pictures, record things that had happened. It's similar to them beginning agricultural pursuits rather than just hunting and gathering all the time. At some point our minds discovered the capacity to create and enjoy something like cave paintings.

Though humans may have started out the same as other beasts of the wild kingdom, we developed differently. We began to think and feel in the way that makes us so different from every other creature living on this earth.

like what Meg said, it is another form of expression. Back then people didn't have a definite language or alphabet. So better way to express what you are thinking or feeling with pictures? It's kinda like asking why we take pictures of ourselves and things.

because they love art