The most lucid answer to this question is the original half-page deduction of the distribution by Maxwell himself which can be found in his collected papers or, more accessibly, in Hawking's anthology "On the Shoulders of Giants". He deduces the form of the probability law from the physical assumptions
a) the probabilities of the components of the molecular velocities in the three coordinate directions are statistically independent;
b) the probablity depends only on the speed, i.e., the absolute value of the velocity vector, and not on its direction.
Mathematically, this means that the only functions of $r$ which split as the product of functions of $x$, $y$ and $z$ are those of the form
$a \exp(br^2)$. This leads to what we now call the normal or Gaussian distribution.