How does SMAP detect soil statistics?
The SMAP includes both an active L-band synthetic aperture radar and a passive microwave radiometer. 

Any object above a temperature of absolute 0 will emit thermal radiation over a wide range of frequencies.  The spectrum of this radiation can vary from that of a theoretical "black body" depending on its physical properties, especially the dielectric constant.  

Since the bulk dielectric constant of the soil depends on the soil moisture, the spectrum of the thermal radiation from the soil is related to the soil moisture.  The dielectric constant of frozen water is different than for liquid water, so there's a different signal depending on whether the soil moisture is liquid water or ice.  Needless to say, it takes a very sensitive and accurate instrument to measure this thermal radiation in order to estimate the soil moisture.  

Similarly, for the active radar, the signal reflected from the soil depends on the dielectric constant of the soil, which is directly related to the soil water content.        

The data from these instruments are combined to produce a surface soil moisture product- this is an estimate of the moisture in the top 5 cm (2 inches) of the soil.  It's important to understand that this is not an estimate of (and is a very poor substitute for) the root zone soil moisture available to crops or other vegetation.  This distinction is important in identifying drought conditions and in predicting transpiration from plants.    

There's a large research literature on using passive radiometers and active radar to estimate surface soil moisture. Is there something more specific about this you'd like to understand?