Why do some animals need to eat green plants?

Answer 1

Their digestive system cannot tolerate complex proteins.

Answer 2

Either you are asking why a critter, like cats, eats grass despite being a carnivore or you are asking why herbivory evolved. Cats may eat grass as a source of vitamins they do not make for themselves nor acquire in adequate amounts in their diet or they may do it to induce vomiting. Wild carnivores get plant material from the stomach contents of the prey they eat or the dung they eat. Dogs frequently eat horse manure on farms. Whales eat herbivorous fish.

All creatures need an energy source. They can be autotrophs like plants and feed directly from the sun or similar energy source or they can be heterotrophs and eat another organism. Heterotrophs specialize in which other organisms they eat. Herbivores eat autotrophs.
Herbivory evolved as a strategy to prey on plants to steal the energy the plants stored for their own use. Plants being sessile are fairly easy to find but are very hard to digest so symbiotic microbes do the actual work. An herbivores strategy is to deal with the cellulose and lignin, found in plant cell walls separately from protein, fat, and carbohydrates. Herbivorous organisms have enlarged digestive tracts, often including fermentation chambers separate from enzyme digestion areas.

Non ruminants (humans, pigs and dogs) digest carbohydrates, protein and fat by enzymatic action. Ruminants (cattle, sheep and deer) use bacteria in the fore stomachs to digest fiber by fermentation and use enzymatic digestion in the small intestines to digest complex carbohydrates, protein and fat. A horse falls between a ruminant and non ruminant. Horses digest grain in the fore gut and hay in the hindgut.