To understand this, start with something familiar - 24 hours of clear skies. During the day, the dominant effect is transfer of energy from the hot sun to the cooler Earth, which happens because the atmosphere is largely transparent to sunlight. At night, the reverse happens because outer space is very cold; the extent to which this occurs depends in part on the atmosphere being at least partially transparent to the infrared light the Earth emits by virtue of its temperature. (As you've also probably noticed, it doesn't cool much on cloudy nights, because clouds are not transparent to that infrared light.)
This material has two main features. The first is that it simply reflects most sunlight, and therefore doesn't absorb much energy in the first place. This could be achieved with any good mirror. The novel part lies in its infrared emissions. Ordinary materials emit infrared light across a continuous band of infrared wavelengths, with the exact distribution governed mainly by temperature. Some of that infrared light is at wavelengths the atmosphere absorbs - that's the greenhouse effect. But not all wavelengths are absorbed equally.

What they've engineered is a material that emits less infrared at the wavelengths the atmosphere tends to absorb and more at the wavelengths to which the atmosphere is transparent. This is how they achieve net cooling. It doesn't violate the laws of thermodynamics because energy is still spontaneously flowing from the warmer body (the material) to a colder one (outer space).