This is just saying that the square root is also symmetric.
For a matrix $A$, there may exist more than one matrix $Q$ such that $QQ = A$. Not all $Q$ are symmetric, but if there is a $Q$ that is symmetric, then so must be $A$, since $A^T = Q^TQ^T = QQ = A$.
One of the easy ways to find $Q$ is to use the eigendecomposition of $A$.
If $Q$ needs to be a real matrix, then $A$ has to be real and positive semidefinite.
In the case that $A$ is positive semidefinite, you can take powers of $A$ to get the inverse, or the inverse square root, etc., like for $R$ in your example.