Unfortunately the answer to all questions of the form “Can you explain [some phenomenon that we observe on a psychological/experiential level] on a neurological level?” is “no”. We’re pre-copernican in our understanding of the mind/brain and all we can do today is say things like “well, activity in x region is associated with y phenomenon”. Which is like asking what “war” is and getting an answer that there’s statistically more physical heat in regions where war is occurring.
But we do have lots of perspectives that you might find helpful or interesting. One that is applicable here is multi-agent models of mind. For example, there might be at the same time a part of me that wants a cheeseburger, and a part that wants to eat healthily. In that kind of picture, cognitive dissonance would be characterized in the same way you’d characterize dissonance between two agents in any system. More about multi-agent models of mind here.