What is carbon in the soil?
It refers to organic carbon, i.e. the carbon present in proteins, carbohydrates and other molecules as consequence of the metabolism of living creatures. Carbon rich soil is an indication that microorganisms, insects, and worms are living in the soil. They are the basis for the soil ecosystem and boost the flow of nutrients that ultimately will support plant growth. A soil with low content of organic carbon indicates that it's a harsh environment where living organisms struggle.

Some articles on the matter:

https://en.wikipedia.org/wiki/Soil_microbiology
http://www.nature.com/nature/journal/v501/n7468_supp/full/501S18a.html
http://mbio.asm.org/content/6/2/e02527-14.full