tensor rotation

You are considering the transformation law of the tensors and this depends on the nature of the tensor. Vectors transform in a certain way and other objects transform in other ways. The transformation of the $T$ you are talking about can be understood as follows.

Consider rotating a vector $v$ by $R$ $$ v^{'}=Rv $$

The operator $T$ maps $v$ to $Tv$.

In the rotated frame the rotated operator $T^{'}$ maps $v^{'}$ to $T^{'}v^{'}$

The mapping $v \to Tv$ can also be achieved via a different pathway i.e. by transforming to the rotated frame and then back again.

Step 1. Rotate the vector $v$ to give $Rv$

Step 2. Apply $T^{'}$ to the rotated vector, giving $T^{'}Rv$

Step 3. Rotate back to the original frame. This needs $R^{-1}$, giving $R^{-1}T^{'}Rv$

This has shown $$ Tv = R^{-1}T^{'}Rv $$

from which follow

$$ T = R^{-1}T^{'}R $$ and $$ RTR^{-1} = T^{'} $$