The direction of angular velocity is different from that of regular velocity for (arguably) two reasons. First, it points out of the plane because of the nature of angular velocity. It signifies a rotation, as such, there is not any particular direction unit vector in every coordinate space that could represent it. In spherical or cylindrical coordinates, it would of course be easy to assign it to the $\hat\theta$ direction, but what about systems like Cartesian coordinates? Thus, to signify the direction of something that points in every direction on a plane, it is easy to specify it along the one direction we can be sure the velocity isn't pointing - normal to the plane. This is a much used convention (such as with area vectors, torque, and many others). As usual as well, we use the Right Hand Rule.
The second, and perhaps more important reason is that we always want to ensure that the angular velocity does not correspond to any true velocity that would be moving in a radial direction. However, to convert angular velocity to true velocity, it is necessary to multiply by the radius (for the most part). Therefore, the equation:
$$\vec v=\vec\omega\times\vec r$$
is used. This allows us to define it in such a way that the true velocity never has a radial component due to the angular velocity.