Let's start with a dictionary definition of "commodity", which is "In economics, a commodity is the generic term for any marketable item produced to satisfy wants or needs. Economic commodities comprise goods and services".
In Qatar's case, the commodity is natural gas, and Qatar is "moving" towards a selling a higher proportion of that "commodity", compared to its other commodity of oil [whose market is dominated by Saudi Arabia]. Simply, "to move to a commodity" is to produce or sell more of a particular good or service than previously.
"Fits with", in this case, could be stated as "corresponds to", or "is in line with", or "agrees with", or "helps".
"That" refers to everything which comes before it in the sentence.