Partition function, by definition can be written,
\begin{align}
Q&=\frac{1}{N!h^{3N}}\int_{-\infty}^{\infty}d\textbf{P}\int d\textbf{R} e^{-\beta U(\textbf{P},\textbf{R})}.
\end{align}
In most cases, the integral over $\textbf{P}$ can be calculated analytically, leaving only $\textbf{R}$ contribution left inside the integral. It is easy to see that direct integration of the Boltzmann factor($e^{-\beta U(\textbf{R})}$), is impractical; you need to sample all possible configurations. Roughly speaking, the partition function in statistical mechanics is like wave function in quantum mechanics. If you know analytically form of the functions, you know everything about the system, but the function itself does not have clear meanings. Of course, knowing the analytical form of the functions is out of reach except for few simplest examples. That brings us to your question of what practical use the partition function has.
My answer to that would be "free energy".
Free energy not only directs in which direction a system will evolve over time but also relative probabilities of different states of the system when it reached equilibrium. For example, under fixed number of particles, volume, and temperature, system's partition function, called canonical partition function is related to Helmholtz free energy, $A$,
\begin{align}
A&=-\frac{1}{\beta} \ln Q.
\end{align}
I am not going to prove the equation above, but maybe you can look it up somewhere online. What I want to show is how such relation connects free energy differences between states to relative probabilities between the states when the system has reached equilibrium. First of all, the range of integral over $\textbf{R}$ in $Q$ is considered along the volume of the system but can be limited to the states of interest. For instance, consider two states, $a$ and $b$, where all particles are limited in $R_a$ or $R_b$ for each state, where both of them are subspaces of $R$. The free energy of state $a(b)$ can be written,
\begin{align}
\label{eq:twopartitionfuctions}
A_{a(b)}&=-\frac{1}{\beta} \ln Q_{a(b)}\\
\text{where, }Q_{a(b)} &=\frac{1}{N!h^{3N}}\int_{-\infty}^{\infty}d\textbf{P}\int _{a(b)}d\textbf{R} e^{-\beta U(\textbf{P},\textbf{R})}.
\end{align}
The range of integral for position vector $\textbf{R}$, is limited to each state $a$ or $b$. The difference between the two free energies can be written,
\begin{align}
\label{eq:twopartitionfuctions1}
\Delta A&=A_{b}-A_{a}\\
\label{eq:twopartitionfuctions2}
&=-\frac{1}{\beta} \ln \frac{Q_{b}}{Q_{a}}\\
\label{eq:twopartitionfuctions3}
&=-\frac{1}{\beta} \ln \frac{\frac{Q_{b}}{Q}}{\frac{Q_{a}}{Q}}.
\end{align}
The denominator inside the logarithm, $\frac{Q_{b}}{Q}$ can be expanded,
\begin{align}
\label{eq:twopartitionfuctions4}
\frac{Q_{b}}{Q}&=\int _{b}d\textbf{R} \frac{e^{-\beta V(\textbf{R})}}{\int d\textbf{R} e^{-\beta V(\textbf{R})}}\\
\label{eq:twopartitionfuctions5}
&=\int _{b}d\textbf{R} P(\textbf{R} )\\
\label{eq:twopartitionfuctions6}
&=P_b,
\end{align}
where $P(\textbf{R})$ is Boltzmann probability, and $P_b$ is probability of system to be at state $b$ in equilibrium. Using the result, the free energy difference between state $A$ and $B$, $\Delta A$ is related to there relative probabilities in equilibrium,
\begin{align}
\label{eq:twopartitionfuctions7}
\Delta A&=-\frac{1}{\beta} \ln \frac{P_{b}}{P_{a}}.
\end{align}
In summary, partition function is related to free energy and free energy is related to equilibrium probability. Knowing the probability is critical on understanding a physical or chemical phenomenon like in the example of drug binding. When designing a drug, we want the drug to favor the target site most, but not other sites in the target protein and sites in other proteins. The probability of the drug binding to many possible binding sites will tell if the drug is a good candidate or not. That is just one example of why free energy calculations are important topics in the field of computational chemistry.
