The vector nature of Angular Momentum

Here is an animation from Wikipedia : http://en.wikipedia.org/wiki/File:Torque_animation.gif

The angular momentum is given by the Cross product of r and p
We can see that the direction would be perpendicular to the direction of rotation of the particle (as shown in the animation)

I don't think this really makes sense, how is the vector nature of angular momentum justified ?

How can one get an intuitive sense about the direction of angular momentum ?