Q:

why laser beams don't get reflected (or knocked away) when they intersect with each other? 

laser beams are photons with the same frequency and the same direction, but according to the wave-particle duality, photons have mass.    
but if we shoot two masses and they intersect at some point they don't go through each other, they get reflected (their direction changes) and might lose some mass.    
but that's not the case with laser beams, if we hold two laser beams and make them intersect at a point, they would just go through each other, neither they change direction nor they produce a new color (their masses and frequencies are the same) because mass is related to frequency from Einstein equation.    
E=hf
m=E/c^2
m=hf/c^2   
why does this happen?  

A:

For two particles to influence each other you need some sort of interaction. 
For (macroscopic) mass this is clearly Coulomb-interaction.  Two atoms can not be at the same place, because their cores repell each other.  If you look at smaller scales, strong and weak interaction might add their part. 
Photons have no charge, no color-charge and don't interact weakly.  So there is no way for them to interact directly with each other.  It might be that gravitation has an effect, but one can easily assume that it is negligible. ( A picture of two people standing on different sides of a soccer-field, shooting with a gun at a 90° angle to each other comes to my mind.  Then assume the bullets don't hit each other but only attract each other via gravitation and calculate the effect.  Might not be in any way a properly scaled example, but it should give an idea). 
However if you go even deeper into particle physics, you will see, that photons indeed can interact with each other via higher order processes.  See e.g. this website. 
As far as i remember from last semesters particle physics lecture, photon-photon-scattering has indeed been observed, but the cross-section (basically: "how often does this happen?")  is quite a few orders of magnitude to low to actually see the effect with your eyes. 
(Think of the two people with guns at the soccer-field again - how likely is it that their bullets actually hit each other?)