I think it's because the light gets scattered by suspended particles that the beam passes through - hence if you shine a laser through smoke particles, you can see the beam clearly.
Unless I am mistaken, what we see to the side is refracted/reflected light which interacts with the molecules/atoms contained in our atmosphere, particularly dust and moisture. It would then follow that a laser beam passing through a total vacuum is totally invisible except in the direction toward the light source (which could seriously damage eyesight).
in the early part of science education in schools you will be taught that light only travels in straight lines, or rays. If you take A level physics, you will learn that light actually displays evidence of travelling in waves (i.e. it can be defracted - see Huygens) so the fact that you can see a laser beam from the side is just another another explaination of this duality, although yes, most of that light is being reflected off particles in the air.
The best way to make lasers visible from the side is to project them through jelly. Yellow jelly for red lasers works best. Prisms or other objects may be placed in the tub of jelly to demonstrate refraction as the beam passes through the denser material.