Like you say, speciation is one of the key points of contention with evolution, because it’s never been directly observed. That, however, is probably due to the extreme time span which is required for it to happen. Simpler adaption is not in dispute. But speciation is just adaption over a much longer timeframe. When two members of the same species adapt differently under separate environments to the point where they can no longer breed with one another, speciation occurs.
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.