Why does the airplane not 'feel' the crosswind?
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.