Let's start how lasers work.
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The term "laser" originated as an acronym for "Light Amplification by Stimulated Emission of Radiation".
How do lasers differ mainly from a natural lightsource? Lasers emit light coherently.
There is basically two types of coherence of lasers:
spatial coherence, meaning lasers are point-like sources
temporal coherence, this means that it implies a polarized wave at a single frequency, that has a single phase over great distances, in other words, the phases describing the mathematical formula of the wavefront are not random.
So to your question, lasers can emit light that goes straight (or parallel beams) over a great distance.
Now in your case, these lasers have to come from mainly one source, the projector, and end up over a bigger spacial area, on the screen. This will mean that the different beams will go not-parallelly towards the screen, but the beams themselves will not spread (they will stay laserbeams). But you are right, the laser beams should not go onto your glasses from inside, like on your picture with normal light.
Why are lasers sometimes visible from the side? The better question is, why is laser not visible from the side in air? It is because lasers are specifically made so that the laser stays collimated, meaning parallel over great distances. It would not be enough, though. Why does laser not scatter on air?
If laser would scatter on air molecules, we could see its light from the side. And it does scatter. It is called Rayleigh scattering.
you could never see a laser from side in vacuum (space), because there is no air to scatter off
in air, if it has dust or anything like that, lasers will be easily visible from the side, as laser will scatter off the dust molecules
even in clean air, sometimes lasers can be visible from the side, this is called Rayleigh scattering, it has a strong inverse dependence on wavelength, so it is easier to see green or blue lasers.