A tensor is independent of coordinates, but its components depend on the basis used to describe it.

Suppose an orthonormal basis is rotated by a matrix $Q$. A vector with components $\mathbf v$ is represented in the rotated basis by

$$\mathbf v'=Q^T\mathbf v$$

under a common passive-coordinate convention.

For a second-order tensor represented by matrix $T$, the corresponding components transform as

$$T'=Q^T TQ.$$

Why two factors appear

A second-order tensor interacts with two vector-like directions. Changing the basis must therefore transform both of those component directions rather than multiplying the matrix on only one side.

Components are coordinate-dependent

A tensor can be diagonal in one basis and have off-diagonal components in another. This does not mean the physical material or stress state changed; only its coordinate description did.

Principal directions

For a symmetric second-order tensor, eigenvectors define orthogonal principal directions. In that basis the tensor matrix is diagonal and the eigenvalues are the principal values.

This connects tensor transformation directly with change of basis, eigenvectors and the principal-stress ideas used in mechanics.