Why linear transformation can improve classification accuracy when the dimensionality of data is high?

Let $X$ be an $m\times n$ ($m$: number of records, and $n$: number of attributes) dataset. When the number of attributes $n$ is large and the dataset $X$ is noisy, classification gets more complicated and the classification accuracy decreases. One way to over come this problem is to use linear transformation, i.e., perform classification on $Y=XR$, where $R$ is an $n\times p$ matrix, and $p\leq n$. I was wondering how linear transformation simplifies classification? and why classification accuracy increases if we do classification on the transformed data $Y$ when $X$ is noisy?

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.