Ostensibly, the purpose of a rudder is to yaw an aircraft. To do this, the control surface on the rudder moves to create lift in the opposite direction of the intended yaw.

Look at the point at which the rudder develops this force. The yawing force is typically centered above the centerline of the aircraft. Well, if it is above the centerline of the aircraft, it's not too different compared to how an aileron works on the main wing. Basically, it is force at a distance... torque... which tends to roll the aircraft in the direction of the lift force.

As for pitch coupling.... induced drag increases when lift increases. It's a similar phenomenon as the above. When you apply rudder, you get increased lift on the rudder with it's associated increased drag. This drag force is above the centerline of the aircraft and points to the rear. Think of it like a shift lever. It pitches the aircraft up... in this case, it doesn't matter whether the rudder deflection is left or right.

This is a simplistic explanation as there are more interference and coupling effects that can take place.