The intention behind negative interest rates is twofold: First, it makes your country's currency cheap by comparison. Who wants to hold a currency that only loses value? Cheap currency fuels exports and drives inflationary economic growth. Second, it punishes people who keep cash in the bank. Rather than let yourself be charged interest for storing it in an account, the thinking is you would rather take it out and spend it, thus generating economic activity.
But negative rates also create a surreal world in which you are charged for lending money and you make money by borrowing it. This chart from the Economist shows just how far negative yields have permeated the government bond market.