Making sense of Haar transform and inverse Haar transform
I fully agree with your option: I consider that "to learn about something, you should teach it first". The Haar transform is important at several levels, here are a few ones:


As an orthogonal basis, it was thought to provide useful (in some strict mathematical sense) expansions of functions
As a 2D dimensional discrete operator, it emulates an oriented singularity detector, allowing to reveal horizontal and vertical orientations
As a regularity absorber, it deals well with locally linear variations
As a multiscale device, it provides an invertible way of looking at data in a multiresolution power-of-two scheme
As an algorithmic tool, it mostly requires adds and subtracts, hence it is very fast and cheap to compute.


On only one level, the result looks as below  

The original image is reversibly cast to a lower resolution version, and three panels exhibiting singularities. They are quite grayish, showing that locally, pixels don't vary that much, except around edges. And you can iterate this again and again.

For locally smooth images with edges, (opinion) it is the most natural idea to represent, detect and compress information at different levels, with very low cost.