You need to learn more, because it does indeed split the species(es) into hundreds (or more) of different "species" over time - however, we do not generally refer to such as species because they are very tiny changes, many of which will breed back out of the population. However, some mutations tends to accumulate because they provide an advantage (or possibly disadvantage, but there is a second mutation that provides a bigger advantage and so both are perpetuated - this is possible to see in some species today, eg: the peacock - its tail provides a disadvantage, but apparently the female peacocks dig those fancy tails so the ones with the best tails get the chicks to breed with - its an example of a runaway mutation).
If enough within the population receive the mutation and it becomes dominant, this is a step along the path to becoming a new species. Depending on the size of the mutation it may or may not get classified by brainy people as a new species or not, it may take many generations of the change becoming more profound or multiple mutations before it becomes recognized as being a new species.
For this to happen there often also has to be some element of isolation of the group so that the mutation doesn't spread to other populations, or other populations do not interbreed with the new variant and wipe out the new feature.
Anyway, you may then end up with two separate distinct species, or more. Most mutations though do not provide any advantage, so they simply die out. So unless you want to go down the road of classifying every change of DNA as a mutation - which would lead to every single human being classed as a separate species, since all our DNAs are unique - then usually it takes some noticeable difference.
The isolation aspect also comes in with hybrid vigour, which can happen both naturally and artificially.