Mr. Putin was referring to a physics phenomenon first described by Scottish scientist Robert Brown in his 1827 paper "A brief account of microscopical observations made on the particles contained in the pollen of plants."
While observing pollen grains suspended in water, Mr. Brown found that the grains jiggled and bounced within the fluid in what he called "rapid oscillatory motion." He could not see any cause of their motion – they didn't appear to impact any other particles, and he ruled out currents within the water – but he was able use other substances to consistently replicate the phenomenon, which he dubbed "Brownian motion."
It wasn't until 1905 that the phenomenon was explained, by none other than Albert Einstein. The grains weren't jerking randomly in the water, he wrote, but rather were colliding with much smaller water molecules. As the molecules hit the grains, the grains would change direction, but since the water molecules were invisible, it would appear that the grains were jittering without cause. Einstein's discovery was seen as proof of the atomic theory of matter: that all matter is made of tiny particles, or atoms.
So what did Putin mean? Presumably, he was referring to the protesters acting without apparent purpose – he also said that their proposed individual programs were not "unified," and that their goals "are also not clear."
But there's also a bit of a warning in Putin's metaphor. After all, the point of Brownian motion is that it's not as inexplicable as it seems – it's only inexplicable because the observer can't see all the molecules in play. Perhaps the same is true of Putin's perspective on the protesters.