This is an interesting question -- "why" questions are very difficult to answer. While there are several levels of "why" a person could ask, ranging from existential/philosophical to mechanistic, I think we can probably answer the "why didn't grass go extinct" question. It seems that grass evolved about 65 million years ago, proliferating mostly after the Cenozoic period, which is when the mass extinction of dinosaurs is thought to have occurred. Grasses are hard to chew and digest for many animals, which suggests at least from an evolutionary standpoint why they flourished: the many herbivores that were left over after the mass extinction would have a tough time eating them.

Grass is a plant (actually, it's a family of plants). Plants grow where plants grow. Grass, being an herb (a non-woody plant), tends not to live where it can get shaded by trees, so you generally get grasses mainly in areas where the trees are sparse (there are grasses that do tolerate shade, but most do better in sun).