Imagine it's a clear day and the sun is shining down upon the Earth.
Let's pretend that the line containing vector v is the ground.
Let's pretend that vector u is a stick with one endpoint on the ground and one endpoint in the air.
Since the sun is shining brightly, vector u would therefore cast a shadow on the ground, no?
The projection of u onto v is another vector that is parallel to v and has a length equal to what vector u's shadow would be (if it were cast onto the ground).