Changes that animals in a species go through don't happen overnight, but they do happen in the span of a single generation. The evolved animal is technically different from its parent, but it still the same species. It replaces the previous version of the species, much like a software update would.
With the main goal being 'survival', some of these changes help the goal (and survive over generations), and some of them don't help survival at all (and fade away from the species over time). After hundreds of generations, the species won't be the same, but it also won't be a new species, just a survivor from the same species as hundreds of years ago.
When Darwin's Theory was first published, it was the Bible and not fossil data that was widely seen as the authority on the Earth's age. The Earth was thought to be just 6,000 years old in the 19th century, which Darwin thought was too little time for all the diverse species on Earth to come to be.
But just years later, in the 1860s, geologists and astronomers came up with a scientific estimate: the Earth is somewhere between tens and a hundred million years old. To Darwin, this seemed like a reasonable amount of time for life on Earth to evolve to its current form. We know today that those calculations of Earth's age are off by 4.5 billion years. The Earth is roughly 4.6 billion years old based on research since Darwin's time.
The diversity of flora and fauna on Earth today is because along the course of history, many new species sprouted from related one. This new species will have unique traits that make it different from its relatives. This is particularly common in one or more groups of animals that get separated or isolated from the rest of their herd for many generations.
The same species under different conditions will be exposed to different environments and adapt accordingly. So, while an elephant in Africa has larger ears so it loses more heat from its body surface, Indian elephants don't have this adaptation. At some point in their evolution, the two separated clans will have enough differences to become incompatible. They will no longer be able to breed with each other as their ancestors once did.
Though Darwin's Theory about evolution and natural selection was spot-on about most things, he followed it up with a theory on "Pangenesis". Pangenesis tries to explain the variation in traits from one individual to the next in a single species.
Darwin proposed that cells produced seeds, called 'gemmules' that are provided by each parent. The gemmules would fuse to make an egg, which grows into a healthy functioning organism if all is well in parents' gemmules. A bad gemmule means birth defects, bad organs. He used gemmules to also explain the strength of one parent's traits over another, but no one since has managed to prove his theory.
One of the hallmarks of good theory construction is the use of separate lines of evidence as proof. To lend support to his theory of natural selection, Darwin took examples from biogeography, paleontology, embryology, and morphology. He noted several examples of “closely-allied species” (that is, closely related species that likely descended or branched off from a common parent species) inhabiting the same territory or adjacent territories. He noted that different zebra species were found together on the plains of East Africa and, in perhaps his most famous example, that several living species of Galapagos finches co-occurred in the Galapagos Islands—a cluster of isolated islands in the eastern Pacific Ocean. The pattern of such closely related species in space supported the idea that these species had a similar origin.