Just as the above explanations has it, my understanding is that dropping flaps adds to the available lift that can be tapped for adding to the rudder just as a higher gross weight does this as well. The idea some people miss is that slipstream quality behind the operating engine is also affected which reduces the thrust on the operating engine and helps reduce VMC that way. I would say the latter factor is the stronger but I doubt anyone knows for sure. You would have to rig out the airplane with a ton of test equipment to really nail it down and no one cares that much.

The OP seemed to be saying in his post that flaps may add to the problem rather than helping it, which I toyed with for a while but dismissed. Some theory I can cook up if it were true, and I do not think it is true, is that having flaps down produces an unequal extra lift factor on the operating side. The problem with this idea is that while some extra lift does happen on the operating side, the induced drag from the flap also increases exponentially with the extra lift, and I think this would be a negating effect.