The theory is that negative (real or nominal) interest rates serve as a continuation of the general purpose of lowering interest rates. That is, they encourage borrowing and spending, both increasing the size of the money supply and especially its velocity. The net effect is supposed to be increased economic activity and higher nominal prices for goods and services. The essential premise is that negative deposit rates encourage people to spend money by in effect confiscating it if they do not. It's not obvious that negative nominal rates are any more effective at this than the negative real rates that have been imposed in much of the world for the last 7 years. Perhaps someone could point us at research on that particular topic, if any has been done.
There are a number of problems with this. Some of them have been mentioned in passing in other comments, and most of them boil down to the disconnect between economic models of humans and real humans. Among other discrepancies, real humans have finite life spans and therefore a limited tolerance for variance; as such, under less than total economic security they generally refuse to dispense with saving if it's an option for them, even if the expected value of spending their savings is greater than not. So the real effect of progressively lower interest rates on monetary velocity diminishes as rates decline. A similar effect limits the ability of lower rates to increase the size of the money supply; people convinced that capital cannot be usefully employed in the present economic environment are unlikely to borrow regardless of how low rates are. A stagnant, highly unequal distribution of capital also limits the effects of lower rates, since the only people likely to be able to take advantage of them already have plenty of capital available to invest. Of course, the money supply side of this is somewhat less relevant, since no one is (yet) proposing negative lending rates, only negative deposit rates. However, negative deposit rates can actually shrink the supply of money, because the "confiscated" money ends up at the central bank where it would simply add to an already-gigantic pool of reserves that could be lent, but haven't been for the above reasons. That is, the money would essentially be removed from circulation and destroyed, potentially offsetting or even dominating any increase in velocity.
You can learn more about the relationships among these quantities by looking up "modern central banking theory" and basic macroeconomic theory in general.
If you destroy cash the idea is to motivate people to invest their cash to get the economy moving. Punish the cash hoarders.
Quantitative easing has not resulted in increasing credit, and therefore the velocity of cash in the economy. If you make it painful to keep cash in one place they hope it is like a game of hot potato and the economy will move again.
The countering strategy is that there is nothing to invest in so it is better to create alternates to cash to preserve value
There are a number of problems with this. Some of them have been mentioned in passing in other comments, and most of them boil down to the disconnect between economic models of humans and real humans. Among other discrepancies, real humans have finite life spans and therefore a limited tolerance for variance; as such, under less than total economic security they generally refuse to dispense with saving if it's an option for them, even if the expected value of spending their savings is greater than not. So the real effect of progressively lower interest rates on monetary velocity diminishes as rates decline. A similar effect limits the ability of lower rates to increase the size of the money supply; people convinced that capital cannot be usefully employed in the present economic environment are unlikely to borrow regardless of how low rates are. A stagnant, highly unequal distribution of capital also limits the effects of lower rates, since the only people likely to be able to take advantage of them already have plenty of capital available to invest. Of course, the money supply side of this is somewhat less relevant, since no one is (yet) proposing negative lending rates, only negative deposit rates. However, negative deposit rates can actually shrink the supply of money, because the "confiscated" money ends up at the central bank where it would simply add to an already-gigantic pool of reserves that could be lent, but haven't been for the above reasons. That is, the money would essentially be removed from circulation and destroyed, potentially offsetting or even dominating any increase in velocity.
You can learn more about the relationships among these quantities by looking up "modern central banking theory" and basic macroeconomic theory in general.