The reason you don't understand how speciation could work is due to your own lack of knowledge.
For example you say that if a new species is born it should die off because it has no one to mate with. This assumes that one species suddenly gave birth to a completely different species. This is not the case. New species are the result of gradual changes over massive periods of time. These changes occur gradually, so there is not a reproductive barrier set up between this individual and the rest of the species because there are not enough genetic differences between it and the rest of the population for it to qualify as a new species. Over time these differences will increase and eventually reproductive barriers will form. However, by this point there will be many other individuals within the new species, so reproduction can still occur.
Sorry if this isn't the best explanation, but this can be hard to explain.
For example you say that if a new species is born it should die off because it has no one to mate with. This assumes that one species suddenly gave birth to a completely different species. This is not the case. New species are the result of gradual changes over massive periods of time. These changes occur gradually, so there is not a reproductive barrier set up between this individual and the rest of the species because there are not enough genetic differences between it and the rest of the population for it to qualify as a new species. Over time these differences will increase and eventually reproductive barriers will form. However, by this point there will be many other individuals within the new species, so reproduction can still occur.
Sorry if this isn't the best explanation, but this can be hard to explain.