Diffusion Capacitance: Forward Bias
Well, there are several ways to look at this question, you have some simple relationships that are developed probably much earlier in a chapter on P-N junctions. From the equations you have we can look at the parameters in the final equation and try and make sense of what is going on. Using the relation that \$I = \frac{dQ}{dt}\$, looking at the final expression arrived at in your question:
\begin{equation}
C_{diff} = \frac{dQ_p}{dV} = \frac{dQ_p}{dt}\frac{dt}{dV} = I\frac{e\tau}{kT}.
\end{equation}
Using the relationship \$V_T = \frac{kT}{e} \$, where \$V_T\$ is the thermal voltage for a given temperature we see that:
\begin{equation}
\frac{dV}{dt} = \frac{kT}{e\tau} = \frac{V_T}{\tau}.
\end{equation}
Where in \$\tau\$ is generally the forward diffusion time constant for carriers in the junction region. So this derived diffusion capacitance is really just telling you how the current and voltage of the junction change with time.

I hope that showing you how to decompose these equations into the usual relationships will help you later on in looking at other similar derived quantities.