Evolution takes place at the population level, not the individual level. Random genetic changes always happen, and over time, some of them become distributed throughout the population. But if a population is split into two populations of the same species (usually through a geological event), then the accumulation of random genetic changes is different in each population. Over time, those changes cause the two populations to be different and eventually to be reproductively incompatible, at which point they are considered two species.
Yes, you have it. Groups of the same species, somehow separated, and the common gene pool is split in two. Then they go completely separate ways.