I believe the language in the text you quote is quite confusing and unusual (but this may only be my perception due to the books I use).
Consider an ensemble of systems, it is prepared in a way, that a system in the ensemble is in state $\left| n \right>$ with probability $p_n$.
A density matrix is an operator of the form:
$$ \rho = \sum_n \left|n\right> p_n \left<n\right|. $$
Where $\left| n \right>$ is a complete (not necessarily orthogonal) set of states.
If we now consider an observable $R$, and consider the expression (for simplicity let $\left| n \right>$ be orthogonal, but it also works out otherwise):
$$ \text{Tr}(\rho R) = \sum_{m, n} \left<m\middle|n \right> p_n \left<n\middle| R \middle| m \right> = \sum_n p_n \left<n \middle| R \middle|n\right> = \sum_n p_n \left< R \right>_n = \left< R \right>$$
Where we evaluate the trace in the same basis set as chosen to specify the statistical operator. That is, the considered expression evaluates to what we expect of an average (the quantum mechanical average of $R$ in the states averaged over the probability of the states)!