Why can Jewish people not eat meat and dairy products together?

The practice of not eating dairy products with meat comes from the Torah prohibition again cooking a kid (baby goat) in its mother's milk. The exact quote is:

"You shall not boil a young goat in its mother's milk" (Exodus 23:19)

The reason for this prohibition is humanitarian. The practice of cooking a young animal in its mother's milk is cruel and therefore not allowed.

Now, why do Jews not eat ANY meat with dairy when the mitzvah is so specific? In Jewish practice, there is a tradition called 'building a fence around the Torah'. What this means, is that, in the Torah, there are many mitzvot, like the one quoted above that are specific. However, there is the possibility that a person can accidentally violate or appear to violate the mitzvah in question.

In regard to the prohibition against cooking a young goat in it's mother's milk, once meat is prepared, it's hard or even impossible to tell the difference between different types. Also, there is the possibility that we might accidentally end up eating a young goat that was cooked in its mothers milk thinking that it was a different type of meat. As a result no meats of any kind (including poultry) are cooked in milk so that it doesn't appear that we're violating this mitzvah or that we unknowingly violate the mitzvah.

The fence was taken a step further by prohibiting the combination of ANY meats with ANY dairy products, including cheese. The logic behind this is the same (so that we don't accidentally or appear to have violated the core mitzvah).