The first image, with equation (4.9), is doing something like this:

All points on a given line of latitude are projected to the same horizontal line on the map. Thus, whatever the vertical scaling of the map is at the height of that line, it is the same for all points on that line. (Because otherwise, nearby latitudes would not be horizontal on the map.)

Since all points at the same latitude are subject to the same vertical scaling on the map, in order to preserve areas it follows that the horizontal scaling must also be constant on that line. Therefore, the horizontal scaling of the map at that latitude can be found by looking at any particular point which happens to be convenient.

That’s what equation (4.9) describes: the horizontal scaling of each line of latitude on the map, expressed as a ratio of an arbitrary longitude to a specific reference longitude.