Physically, I like to view it in the following way:
You are looking at an adiabatic evolution of an electron confined to some energy band $n$. You somewhat project out the presence of other states in other bands $m\neq n$. Still these ignored bands effect the dynamics of your adiabatic state.
When other bands are approaching the energy of the adiabatic eigenstate the Berry curvature increases. If they move further away, Berry curvature decreases.
For some intuition on the QHE I recommend Laughlin's Nobel lecture! To summarize (I losely follow the Bohm's book on Geometric Phases): the basic idea is to consider a two-dimensional electron gas (2DEG) on a finite cylinder. Perpendicular to the 2DEG you have a constant magnetic field $B$. Imagine you would change the magnetic flux through the cylinder by $\delta\phi$ . This changes the energy of the system by an amount
$$
\delta U = I \delta \phi,
$$
where $I$ is the induced azimuthal current. If you thread this flux in an adiabatic fashion and choose $\delta\phi$ as the flux quantum $h/e$, the bulk system returns to its initial state. However, in this process the $z$-localization of the eigenstates shifts. This means if you apply an electric voltage $\delta V$ along this direction you change the energy by
$$
\delta U = n e \delta V,
$$
where $n$ is the number of electrons shifted from one edge to the other. Combining the two results you see that
$$
\begin{align}
\sigma &= \frac{I}{\delta V}
= \frac{1}{\delta V} \frac{\delta U}{\delta \phi}
= \frac{ n e}{\delta \phi}
=
\frac{n e^2}{\hbar}.
\end{align}
$$
A more rigorous treatment of this hand-waving argument can be found for example here.
As you know, this integer $n$ is actually a topological invariant of your electronic band structure. It might be easy to calculate the Hall effect for 2DEG's, as for example done by T. Ando et al. in 1975 (see here) predating von Klitzing's discovery of the QHE! However, this leaves many open questions. For example, why is the QHE so ridiculously precise? Von Klitzing's experiments revealed a very high accuracy.
These properties are elucidated by Berry phase physics because of its connection to mathematical theory of Chern classes.