when we run the regression of our portfolio vs the different style classifications and/or a benchmank, what is the unexplained variation telling us? how the managers investemnt style differs from the model paramaters?

The amount of variation that can’t be explained by the model. In this case, exposure to the style factors.

It tells you to what extent the portfolio’s return can be explained by everything EXCEPT for the style factors (indices, systematic risk, beta…whatever you want to call it). Often, you interpret it as the ability of an active manager to add value through his ability to select stocks, i.e. selection skill as opposed to simply holding/replicating the index.