Okay. First things first. Geographical isolation happens all the time. Mountains may rise in the middle of a species' range; a river might split a population in twain; the isthmus of Panama even did a number on quite a few species.
Once the two or more populations no longer have contact with each other, any mutations which arise in one population stay in that particular population -- they do not get shared with the other populations and the populations become increasingly genetically divergent.
None of the groups is "mutating faster than the others" -- they are simply mutating differently. If, for instance, the environment inhabited by the lion's share of the population hasn't significantly changed for a long time, the population isn't likely to be too different from its ancestral population -- but the small group that got stuck in a rather different environment would have to change rapidly or face extinction. In this case, you'd have the mother and daughter species coexisting. Of course, when the population was split, the different subpopulations might find themselves in rather untenable environments and all the populations would be forced to adapt to local conditions or die off. In this case, the mother species disappears and is replaced by one or more daughter species.
The reason speciation occurs more rapidly if the isolated part of the species is small is that the genetic mutations move more rapidly through a small population than a large one, because the individual with the mutation represents a larger chunk of the breeding population.
Now note, none of this is going on in the blink of an eye -- It still takes generations for the genetic differences to render the resulting populations unable to interbreed. Only when looking back through the telescope of time do we get the impression it happened rapidly. The reality is it still took longer than most civilizations have existed to affect the changes that lead to speciation.
The above is just pure supposition and there is proof against it. One great example is the camel and the llama. These two "species" became separated tens of millions of years ago. They developed differently, they adapted to fairly different environments, yet they remained one and the same species throughout this large period of genetic isolation. Just a few years ago they were able mate one with the other. If evolution is that slow in producing new species, then it is totally impossible for it to be the answer of the vast variety of life on earth.
OK, then once again the theory is faced with the problem of the "gappy" fossil record. Where are the countless "micro-transitional" forms?
Another problem remains. According to the theory, at some point one creature in the "daughter population" must mutate sufficiently to become "reproductively isolated" from the "parent population." But at the same time this same creature would become "reproductively isolated" from the rest of the "daughter population," unless another member of the daughter population happened to mutate similarly and simultaneously.