Is there an intuitive explanation of transverse shear stress? I know the formulas and the equations etc. But how can I visualize transverse shear stress in lets say a beam with square cross section?
And what is a physical explanation of why is transverse shear stress is maximum in the neutral plane despite the fact that neutral plane neither experiences stretch nor compression during bending?
 A: Consider a cantilever beam with a square cross section and an end load pointing downward:

If you cut the beam at any point, there must be a transverse distributed load that adds up to $F$ or $-F$ (depending on what side of the cut you're looking at) for static equilibrium to apply.
In addition, note that a true state of shear requires traction forces on all four sides to prevent the infinitesimal element from displacing and prevent it from rotating:

But there can be no such traction force on the top surface of the beam because we haven't applied one. 
The simplest explanation is that the internal transverse distributed load looks like this:

with the maximum value at the center and a value of zero at the top and bottom surfaces.
A: Consider a cantilever beam with a single loading on the free end

Now recognize that as we move from left to right the internal moment decreases, and this causes the stress distribution (red arrows above) to have less slope across the cross-section.
As a result, when only a small part of the cross-section is considered near the surface you can see that the sum of the forces horizontal won't balance out unless there are shear forces acting on the bottom (blue arrows above).
Now if an element has shear acting on one face, it must have an equal amount on an adjacent face, and an equal and opposite about on the opposite face. The diagram in @Chemomechanics answer shows this clearly.
So now we have a situation where the vertical shear forces are a function of the horizontal force imbalance. For each location along the beam, this imbalance depends on the vertical height of the sub-element. 
In the extreme case of the sub-element being super thin and encompassing only the outer fibers of the beam then the imbalance is zero. On the other hand, if the sub-element covers half the cross-section, from the neutral axis to the outer fiber then the imbalance is maximum. 
I hope this helps you visualize the source of the shear stress distribution.
