Why linear transformation can improve classification accuracy when the dimensionality of data is high?
Multiplying by the $n * p$ matrix decreases the dimensionality of the data set.  Think of this as projecting the highly dimensional space into a smaller dimensional space.  For example, you could do principle component analysis and project it into a small space.  This way things that are correlated together are projected into the same dimension and if one of those correlated dimensions is off because of noise, then it would be cancelled out by the values in the other correlated dimensions.

You have to choose your matrix $R$ in a way that essentially averages together things that reinforce each other and reduce noise.  Principle component analysis is one way of choosing such a matrix.