Punctuated equilibrium attempts to explain how animals arose “so quickly” in the fossil record (e.g., the Cambrian explosion). Basically, the modern, conservative view says that large populations of species split up into smaller subpopulations. These sub-populations are where the major evolution takes place. When the population comes back together (or interbreeds), then this new “evolution” that happened in the smaller, subpopulations quickly spreads through the entire population. Because the evolution occurs in smaller subpopulations, we should not expect the fossil record to contain a smooth transition of species but rather to contain “gaps,” which it does.
See also: Problems with punctuated equilibrium