Couple of things.
First, for the models to work well, they use pressure as a vertical coordinate system instead of height. Instead of 2,000 feet 4,000 ft 10,000 ft and so on, you have 900mb 800mb 700mb all the way up to 10 mb. It’ll be too complicated for the models/equations otherwise.
Second, geopotential height patterns tells you where the warm air masses and cold air masses are. The higher the geopotential height at the 500mb pressure surface (think coordinate system), the warmer the airmass. The lower the geopotential height is, the colder the air mass.
Now, when you have a cold and warm airmass butting up against each other, there is wind… and the stronger the temperature gradient between the air masses, the stronger the wind is. And it is that wind that controls the flow and track of storm systems.
Took me three paragraphs to explain it. Maybe someone else can do it in one.