What does the physical meaning of $\mathbf{U} \Sigma \ \mathbf{V}^{*} $ in the MIMO system?
Singular value decomposition (SVD) is a method of decomposing a matrix into two unitary matrices and a non-negative definite matrix of singular values.

In the context of MIMO communications, the $\mathbf{H}$ matrix defines the channel; it doesn't specify anything directly about the transmitter or the receiver.

This paper has the following diagram which shows one interpretation of the $\mathbf{U}$ and $\mathbf{V}$ matrices.  The SVD allows the channel to be decomposed into several parallel paths, with no crosstalk between them.