What was so unlawful about picking and eating grain on the Sabbath?

“...He was passing through some grainfields on a Sabbath; and His disciples were picking the heads of grain, rubbing them in their hands, and eating the grain. But some of the Pharisees said, “Why do you do what is not lawful on the Sabbath?”

So what was so unlawful about picking and eating grain on the Sabbath and why does Jesus answer with a seemingly obscure story from the days of King David? Maybe knowing a bit more about Hillel and Shammai will help us to understand why "some" Pharisees thought such a simple act of eating was so unlawful.