"Magic X" is a common expression for things which aren't adequately explained but that "just work".
For instance, in programming, it is considered poor form to use "magic numbers". These are numbers which simply appear in a given code without any explanation of what they are. For example, if code has
x * 0.707, you immediately start wondering what that 0.707 comes from. You'll need to look at the code and think about it and remember that the cosine of 45 degrees is approximately equal to that.
Instead, one should probably define a named variable (for example
var cos45 = 0.707) with that number as its value. Then you just do
x * cos45 and it's clear as day what the code does, no thinking required.
That being said, if you do come across a program which does
x * 0.707, you might not really understand where that number comes from, but you can tell it "just works".
In a similar vein, "magic formula" simply means it's a formula which, for reasons unknown, "just works". That much is stated in the Wikipedia article you linked to (emphasis added):
Pacejka has developed a series of tire design models over the last 20 years. They were named the 'magic formula' because there is no particular physical basis for the structure of the equations chosen, but they fit a wide variety of tire constructions and operating conditions.
One of the references to the Wikipedia article even keeps the symbolism going (emphasis added):
The next step is to compute the product of a new helper, $B$, times $b_0$ and the aforecomputed $D$. The magicians who created the formula tell us that $B b_0 D = (b_3 F_z^2 + b_4 F_z)\exp(-b_5F_z)$. This slurps up a few more of the magical eleven empirical numbers, and a pattern emerges [...]
Basically, the "magic formula" is so-called because it is totally empirical. There are many other empirical formulas in life, though, so you might be asking why "only" this one is called thus. The answer is that this one is really empirical. Other empirical equations are mostly intuitive: they may have empirically-determined coefficients, but the overall form of the equation makes some sense. This one is just a bunch of coefficients added, multiplied and exponentiated and somehow it all works out.
Or, as that same reference I linked to above puts it:
Once again, don't try to find any physics in here: it's just a convenient formula that fits the data reasonably well.