There are several mechanisms by which a new species may arise through the various types of isolation.
First, it's important to consider that the definition of "species" is, in some ways, a concept invented for human convenience, and not an actual biological condition. For example; we would clearly consider horses and bananas to be different species, because they cannot reproduce and pretty much have nothing in common. However, for species that are closely related and can interbreed, even if it's just occasionally, the human concept of species places the condition that the two groups "do not regularly interbreed", which is a very loose term and really has nothing to do with the actual genetic components of speciation. It's like saying that your family isn't really your family if they don't visit often enough.
Isolation is an important factor in speciation because it keeps the genetic pot from being stirred, so to speak. If all of the genetic traits in a population are constantly interacting with the others, then there's no real opportunity for minor variations or mutations to assert themselves against the mass of the genetic body as a whole. For example, if some humans were born with a recessive trait for detecting coconuts, this might be an interesting novelty but it wouldn't initiate a speciation event, because there are simply too many people and too many other competitive traits out there (and you wouldn't really find this to be a compelling reason to have children with someone). However, strand that coconut-detecting person on an island with only a few other people, and suddenly they might become extremely useful and sought-after, greatly increasing their genes as a proportion of the whole population.
Isolation keeps other genes out, but it also keeps the existing genes "in" - that is to say, it amplifies the expression of the genes available within a particular pool. On a large scale, this is evidenced by research into race; Europeans, for example, have far less genetic diversity than continental Africans, and two Africans from opposite sides of the continent are in fact many, many times more genetically distinct than any African is from a European, suggesting that Europeans represent a small but greatly amplified subset of the overall human genome. One way that this kind of condition could lead to speciation is through physical barriers to reproductive interaction; if the isolation takes place on an island, or if the species is being selected for great heights, then it would be physically challenging for closely related species to even attempt reproduction.
First of all, if a species becomes isolated, it has the opportunity for its gene pool to remain constant. There are no other species to interfere with the way the species grows or change the genes in the offspring. Isolation also limits the number of predators that can potential kill off the species, causing rapid population growth. However, that could lead to a limited number of resources in the area. There are pros and cons to isolating a species, but those are some positive effects that could come from it.