The saber-tooth, or Smilodon has always been one of my absolute favorite extinct animals—and one that we likely just missed in North America by a small window. The image above is Smilodon fatalis, the species from North America and the northern part of South America. The larger species, Smilodon populator seems to be only found in the trans-Andean part of South America (which is admittedly, most of the continent.)
Their are a lot of potential explanations for their extinction, but probably one of the most significant contributors was the climate change as the interglacial that we currently are living in started. Much of the available land, which was admittedly more square mileage, also became more arid. Pluvial lakes dried up. Forests turned to savanna and savanna turned into open prairie. This significant ecological shift may have been supplemented by other problems; competition from human hunters being one that's frequently mentioned. Ultimately, this is all somewhat speculative, however, and may require further revision. For example, the extinction of prey is often given as a reason for the extinction of predators that in the past had been able to exploit plenty of different environments, but is this really true? In spite of the dramatic image above of Smilodon preying on a young mammoth, bone isotopes suggest that in the La Brea region, the most common prey was bison and Camelops. Sure; American camels went extinct (with the exception of the llama group in South America) but bison didn't. If Smilodon could hunt Bison antiquus, why not Bison bison? They weren't that different, and up to the historical period, woodland bison (as opposed to the plains bison that are more familiar) are known from as far east as the Eastern Seaboard. Although habitants did change, and the Great Plains would have been a more savanna like environment rather than open grasslands, with an open forest feel, and relatively more trees and brush than today, such environments certainly exist in plenty of large areas of both North and South America; it's not true to suggest that everything became drier and more open, obviously. The whole of two continents hardly turned into wide-open prairie and pampas.
I'm also personally skeptical that the Quaternary extinction event was as sudden and dramatic as is often presented; in which absence of evidence is given as if it were evidence of absence. There are a lot of fragmentary fossils here and there of supposedly extinct megafauna that post-dates the main period of extinction, and I'm less inclined to be dismissive of old injun lore that "grandpa hunted a mammoth" or " we had our own native breeds of horses all along" and whatnot than most. While I agree that the preponderance of such stories isn't strictly speaking a fait accompli in the world of logic and scientific conclusions, it does get to be a little hard to ignore and yet still be taken seriously. Not that they don't anyway, but as I read more and more of these anecdotes and more and more samples of animals that aren't supposed to have been there start to turn up, I wonder why that's not really being addressed.
Anyway, yeah—Smilodon; the saber-toothed tiger. I always laugh when people feel inclined to point out that it's not an actual tiger, or even a pantherine cat at all, but part of the machairodont line; a subfamily with Felidae that split early on. So what? There's a lot of what appears to be spergy autism in pop science books and pop science discussion (this is especially prevalent in cladistics discussions and taxonomy); as near as I can tell, literally nobody ever actually thought that saber-toothed tigers were actual Panthera tigris tigers with big teeth; they were always recognized as just big cats. Most people probably believe that they looked more like lions with big teeth anyway, given that that's the coloring Charles Knight gave them.