Sorry this is long, but you asked for a “good” answer.
Speciation is just one convenient but arbitrary cutoff point in evolution. A general but somewhat flawed definition is that species can breed with one another and produce viable offspring…so if they can’t breed they are different species.
Evolution includes any change in (allele) gene frequencies within a population. So for instance. In Australia they released a virus called myxamatosis in the 1950s to kill rabbits. This killed the vast majority of the population (around 500,000,000!). But not all. Those that survived in the 1950s are the ancestors of the current population, which as you would predict are mostly resistant to myxamatosis. This is evolution without speciation. The current population is a specialized subset of survivors from the populations that occured in the 1950s, but are not a different species.
(SELECTION PRESSURE) So the first point is about selection presure. If something selects for or against some aspect in the population, then the population changes in the frequencies that certain genes occur through the fabouring one some individuals and the failure of others. Their relative difference in reproductive success alters the frequency of genes found in future populations (natural selection). So as environments change, species can change through time. Or if some individuals like the appearance of the opposite sex which has certain distinct features, then the populations can evolve and or diverge through this selection pressure (sexual selection).
(GENERATIONS) Regarding selection pressures and environmental change….If we could shook hands today with each of our ancestors, going back one generation each time, we would eventually shake hands with an individual that was no longer the same species as we are because different selection pressures had changed their genome to the point that we couldn’t reproduce if we wanted to.
(ISOLATION) If isolation occurs that separates one population of the same species from another, then the different environments are likely to push the two separated populations in two different directions. If this gets extreme enough then they may not be able to breed with each other, which ensures that they can diverge more (because they no longer mix their genes). This is why humans did not descend from chimpanzees, because at some point around 6 million years ago humans were not humans, and chimps were not chimps. We both shared a common ancestor that we both descended from. New world monkeys and humans share a common ancestor from about 40 million years ago.
The more generations you go back to find a common ancestor, the more change. This is how we get what we refer to as families or classes or kingdoms etc…(eg. not only can they not interbreed, but they are so different that they require a classification that explains this difference…)
(MUTATION) A mistake during cell division can cause a change in the number of chromosomes. Instantly, the offspring may not be able to breed with the parent, but can breed with each other. In that case you can get a parent to give birth to what we refer to as a new species. Eg. a species of Corydalis (small plant) on Gotland in the Baltic exhibits this mutation…polyploidy.
So the end result is there are over 250 species of monkey around today because they have managed to successfully reproduce and adapt to the variety of different environments they inhabit. There are humans around today because we have done the same. At some point between 60 and 80 million years ago all surviving species of monkey and ape (eg. us) share a common ancestor.
There is nothing in this story which suggests that the existence of one speciesthat prevents their ancestors from successfully producing other lineages. If this was the case, the world would only contain a single species, because all extant species are thought to diverge from a common ancestor at some point in the past. You just have to go back far enough. I hope that helps.
P.S. Being a theist can prevent some people from accepting the evidence for evolution, but accepting the evidence for evolution does not make you an a-theist.