In my algebraic geometry course the higher direct images $R^i f_* \mathcal{F}$ of a sheaf of abelian groups $\mathcal{F}$ on a topological space $X$ were introduced as the right-derived functors of the pushforward $f_*$.
While I have a good intuition of what the pushforward is supposed to do (thinking about pushforwards of vector bundles in differential geometry), I have know idea about how to visualize the higher direct images.
Is there any concept from differential geometry which is analogous to higher direct images or any other interpretation?
The higher right derived functors generalize cohomology. If you take the map $X$ to a point, then the right derived functors are exactly the cohomology of the the global sections functor. In particular, cohomology of the constant sheaf $\mathbb Z$ gives singular/cw/de-rham cohomology.
You should think of the higher derived functors as a way to patch together the cohomology of the fibers $X_y$ for a map $X\to Y$.