Glendalegoon's triple expansion steam engine comparison is the first thing that occurred to me too, however another analogy would be a wing with a multi-stage flap.
Each aerofoil can only deflect air down (thus generating lift) by so much before it experiences separation (stalls). If you want maximum lift (on the wing & flaps) or power extracted from the turbine, you need more stages.
A four stage LP Turbine could be compared to, say, an A321 wing with a three stage flap (along the lines of the bottom image).
As the air leaving the aerofoil ahead is deflected, the following one needs more incidence or pitch to extract lift from the downward moving airflow.
