How is Dark Sky able to predict weather on such short timescales? My impression has always been that weather prediction is fairly imprecise and so it only make sense in larger time scales (i.e. it will rain today, it will rain tonight).
The moisture (gaseous water) in the rising air condenses at altitude due to low temperatures, and hence frees additional heat which increases the upwards vertical motion of the airmass. This causes the cloud to grow. At some point, there is too much condensed water in the cloud and the water falls back down - a rain shower.
The development of Cumulonimbus clouds can be observed on weather radars. I'm guessing that they have an algorithm which, given the known conditions of temperature and humidity computes the time and size a Cb cloud has to be in order for a rain shower to start. The growth rate of a Cb cloud can be inferred given the vertical temperature distribution in the atmosphere.
It could also be that things are much simpler and that precipitation can be predicted from knowing the density of water in a cloud, which is what a weather radar measures. I am not a meteorologist.
Basically, we rely on the fact that while weather is chaotic and turbulent in the long run (by "long run" I mean hours to days), it's fairly linear on the timescale of minutes.
So we're able to extract velocity data from NOAA radar images, and use that to project the storms into the future. Some storms are more coherent than others so we're constantly monitoring how accurate our predictions are and adjusting the projections accordingly.
Sometimes this technique breaks down, but it's surprisingly rare.
The first problem is that generally when talking about weather predictions you're talking about a large area. Asking, "will it rain in New York?" can be a loaded question if you're talking about a scattered storm. My understanding is that a lot of the time when you hear 60% chance of precipitation, the weatherman doesn't mean 40% chance it won't rain in the area, but that roughly 60% of the area or population will see rain. So the first advantage this app has is that it's forecasting for you individually in a very specific location, rather than for an entire TV or newspaper market.
Secondly, I think the short time frame is actually much easier to predict. On a short timescale you basically just look at the radar for the past 15 minutes, and see where a storm if any is heading. I do this all the time with weather.com. Open up the radar, see if a storm is nearby and where it's headed. It seems much harder to predict what a storm will do hours or days out, than what it'll do in 15 minutes.