Why Do Ruminants Require A Multi-Compartment Stomach To Digest Food?
There is more difference than just the parts of plants that are eaten.

Two of their stomachs - rumen and reticulum are not used for digesting food at all. Multiple rounds of chewing and mixing with saliva, wich result in very small particles of undigested food help the bacteria in those stomachs reach, digest and use that cellulose-rich plant material. Cows "eat" the bacteria from their stomachs and the grass is just food for these bacteria. You could say that cows are not herbivores but secondary consumers.

Look at this picture (from Wikipedia article on ruminants, a good place to check all this in more detail but still simpler than most professional books):


See that the ruminants have the same second and third part of their digestive system (though the bacteria use up most of carbohydrates, so some difference in absorbed material), but have an additional part in the front - where the microbes digest and ferment the food, that the ruminant has provived them. The reticulorumen needs to be large in order to provide space for all that plant material and bacteria.