I am at Point A, and I have a laser pointer. I point it at a mirror, one mile west of me, at point B. The beam travels to the mirror and comes back to make a nice round dot on my shirt at point C. In my frame, Points A and C are the same.
Bob zooms over me in his spaceship, going half the speed of light due north. Since he is going half the speed of light, he travels one mile in the time it takes my laser to go two miles. In his frame, he sees my laser fire at point A, hit the mirror at B, and reflect to point C, which is a long ways south of Point A.
As we all know, the Lorentz transformation arranges everything so that both of us see the light beam traveling at speed $c$. But the Lorentz transformation does not change the fact that my laser is pointed due west, in my frame and in Bob's frame. Yet Bob sees the beam travel at an angle, and reflect at the opposite angle, to arrive well south of where it originated.
In my frame, the laser is stationary, and the beam it emits travels parallel to its axis. In Bob's frame, the laser is moving south at half the speed of light. How does that motion cause the beam to be produced at an angle to the laser's axis?