As many have stated before me, you can't just throw electricity away if you produce too much—it has to go somewhere. If you put more power into the system than the resistant (or consumption) in the system, it is like spinning your bike faster while you bike downhill, the generators start spinning faster, and the frequency increase above 50 Hz.
The second problem is that many power plants just can't shut down. Nuclear power plants take days to turn off, coal plants hours. They, therefore, are willing to pay almost infinitely much to avoid having to ramp down production on short notice.
Here is a plot of the bid curves on Nord Pool (the electricity market for Norway, Sweden, Danmark, Finland, Estonia, Lithuania and Latvia) two weeks ago, when the price was especially low, just 4.1 EUR/MWh, for one hour, 4am, at night. (You can find a video/animation of how the bid curves vary in the same period on YouTube.)
You can see that over 22 000 MW (MWh/h) of the power provided are willing to produce power no matter how negative the prices become. Around 10 000 MW of this is nuclear power. The rest is probably coal, some other slow responsive thermal power plants, and some producers that won't be bothered with adjusting their production (small power plants, etc.).
Ca. 10 000 of the power provided, 6200 MW of which is wind, bid around 0 EUR/MWh. Since it is impossible to store energy wind producers and hydropower producers without reservoirs are willing to sell for nothing, however, they can turn off quickly if prices becomes negative. Ca. 30 000 MW of the power provided are regulable power, mostly gas and hydropower from reservoirs, that bid around 25–50 EUR/MWh.
Notice also how insensitive consumers are to prices. Demand is slightly higher if the price becomes negative, but it is almost nothing, only ca. 1000 MW.
In the extreme event that prices would fall below 22 000 MW you can easily see how prices would spike down quite extremely. In the Nordic market(*) where much of the production is regulatable hydropower, this is very unlikely.
Spot price and power so in Germany December 2017
The problem in Germany is two folded. First, they have a lot more nuclear, coal, small power producers and other unregulated power sources, than in the Nordic market. So when energy demand is low and the wind suddenly starts blowing and generates 80% of the power needed, like it did on Christmas eve 2017 (see Chart), you get too much energy.
However, normally you would expect wind power producers to turn off their turbines, when the price becomes negative. The problem is, as I have understood it, that renewable producers are guaranteed a minimum price for each unit of energy produced. So instead of bidding 0 EUR/MWh, they too bid infinite negative prices. It's only when the spot price is negative for more than six hours, renewable power producers are not guaranteed the minimum price anymore.
(*) For some individual price areas in the Nordic market, like Danmark, it is more likely with negative prices. However, data on bid curves for induvidual price areas are not public available.