It's not intuitive that the direction of angular velocity is perpendicular to the plane of rotation of the object? Can someone explain?
Because we decided it should be.
The direction of angular velocity is easy to conceptualize and hard to describe. If something is rotating, it could be rotating clockwise or counterclockwise. So why do we point the angular velocity vector perpendicular to the plane?
Because it's convenient. Vectors can't be curved because they're nothing more than a magnitude and a direction. So we use the direction perpendicular to the plane to convey the rotational direction. Furthermore, equations related to rotational kinematics, dynamics, etc. make much more sense and are greatly simplified when using this convention.
In the same way that the sign of linear velocity represents the direction of motion (negative velocity means moving backwards, not moving with negative speed), the direction of angular velocity represents the direction of rotation.