Why are interest rates negative?

The simple explanation is that the supply of loans (savings) is so big, and the demand for loans (investment) is so small, that the price that equates supply and demand (the interest rate), is negative.  

Japan was the first to reach negative rates, and has three distinctive characteristics:

1. A high saving East Asian culture.
2. A mature, fully developed economy.
3. A population that leveled off, and is now falling. 

Only one of the three applies to China, but in the future all three may apply. If 1.4 billion Chinese become rich, and keep saving a large share of their incomes. I'm not certain there are enough productive investment opportunities to absorb all that saving, at least at the sort of rates of return that we are used to. In my view, the growing Asia-fication of the global economy may put further downward pressure on interest rates over time.

For an alternate explanation see Tyler Cowen's article in Bloomberg.