Lay and Lie

Ladies and gentlemen, 

On your left, a transitive verb (one that takes a direct object, as in "I'm going to lay my burden down," or "We laid a trap for him.") 

On your right, an intransitive verb (which does not take a direct object, as in "Please lie on the couch," or "At Flores in the Azores Sir Richard Grenville lay").

I have never understood what seems to me an exclusively US tendency to substitute the transitive for the intransitive, as in "Come and lay by my side". I always find myself asking, "Come and lay what by my side? An egg?"