Why does my infrared thermometer say the sky is at -2 °C?

You are likety to be directly measuring the greenhouse effect of the atmosphere. The fact that your cursory measurements seem to be correlated with the air temperature around you supports this idea: the measurement is higher in the day because the Earth itself is hotter and radiating back into space more powerfully. Greenhouse gasses are thus absorbing some of this radiation, and sending their re-radiated infrared back to your thermometer. The thermometer will also be influenced by clouds and the like, even mist and haze that may be hard to discern as clouds.

I would try noting the thermometer's reading with different atmospheric conditions, noting the sky condition (cloudy, clear), local temperature and humidity. Although I wouldn't be too surprised if the reading were not influenced by the last: humidity at the ground may not give a reliable idea of how much water there is in the various jetstreams and so forth at different heights.

But you're definitely measuring something "real" and repeatable: see the following experiment, which is a formalisation of what you have done, suggested by NASA:

http://mynasadata.larc.nasa.gov/804-2/1035-2

All bodies above absolute-zero emit some radiation. This is "black-body" radiation and it can be correlated to temperature using the Stefan-Boltzmann law. Your infrared thermometer uses this to calculate the temperature by measuring this radiation.

The temperature you measure for the sky is the radiation of an equivalent blackbody at -2°C. This is sort of an "average" of the temperature of the air column above you (but take into account that the Stefan-Boltzmann law has a power of four in there so it is not a linear average).

I come from a colder climate and I can tell you that snow on the ground will persist when the air temperature is above 0°C for days or even weeks. One reason is that the sky is below the temperature of the snow and the surrounding air. The snow can emit more radiation to the sky than it absorbs, offsetting the heat gained from the surrounding air.