Link Between the Density Operator and the Partition Function and Boltzmann Distribution in Quantum Statistical Mechanics

I have a very limited knowledge of statistical mechanics, but I seem to running into some related concepts for my background readings for the research project this summer. For example, see the expressions for $\rho_{B}$ and $Z_{B}$ between equations $(C.19)$ and $(C.20)$ in the attached image:

I have no idea how one has made the link between the density operator and the partition function and the Boltzman Distribution. I found the following notes (link: http://bohr.physics.berkeley.edu/classes/221/1011/notes/density.pdf) that seem to explain this topic in section 16, but I failed to understand them at all.

I won't be comfortable with statistical mechanics unless I take a full course starting next semester. In the mean time, if anyone could either link me to better explained notes or explain these concepts over here, that'd be great.