Why does the airplane not 'feel' the crosswind? Apologies for multiple pictures but I thought they would depict my question more clearly than a lengthy description.

Image 1: Why does the aircraft still follow the straight path even if the wind is blowing from left to right?

Image 2: Why does it appear that the aircraft is sliding?

Image 3: Why is there no need to apply any rudder in the presence of a cross wind?
When dealing with crosswinds and drift, I find the easiest way to understand the concept is to imagine a fast flowing river of water. If you tried to swim straight across a fast river you will end up downstream. You don't "feel" the current. You just keep swimming and the current carries you.   

Any time you have wind it is like a river of air flowing over the earth.  

Traveling through a river of air is exactly like traveling through a river of water.  You don't go where you are pointed, you go where the current carries you, plus the direction you are swimming.