If the distance to the center is a uniform random number, and the direction is a uniform random number, the distribution would be denser near the center of the circle.

To explain, if we choose the area of interest to be the unit circle, it will be equally likely for a point to land in the circle around the origin of radius 1/2 as it is to land anywhere else in the unit circle. However, the area of the smaller circle is 1/4*pi while the area of the outer shell (the "anywhere else" mentioned earlier) of the unit circle is 3/4*pi, which is not the same.

The only way to get an equal distribution would be to make the random distribution increase linearly from the center of the circle, which involves taking the square root of a uniform random number, and square roots are not cheap.