Let me try to expound a very simple argument.
A 3D element with piecewise linear shape functions can exactly represent a constant strain solution. (Consider e.g. the patch test.) Loosely speaking this means that when the exact solution exhibits a strain gradient, mesh size has to be adapted so that there is not too much change of strain across one single element. This holds true also for hexahedral elements, where some linear terms arise in the strain expression: however linear hex element cannot represent exactly an arbitrary linear strain field.
If we admit that at the limit of very thin slabs the 3D solution approaches the Kirchhoff–Love plate solution, then we have a very peculiar situation: mild strain gradient in the in-plane directions $x_1, x_2$, but a very steep gradient in the transverse (thickness) direction: remember that strain (stress) changes sign across $x_3$.
Now it should be clear that this strain/stress state cannot be sensibly approximated by a single linear FE in the $x_3$ direction.
In other terms, for thin plates (Kirchhoff-Love) the most simple straining solution is constant curvature $\kappa$: no gradient in $x_1, x_2$, but a gradient equal to $\kappa$ along $x_3$ of the in plane longitudinal strains $\varepsilon_{11}$, $\varepsilon_{22}$ and vanishing transverse shear strain $\gamma_{13}=\gamma_{23}=0$. A coarse mesh should be able to represent the "simple straining" state, but this is not possible with linear elements: in fact while, as pointed out by Bill Green, it is possible to have $\varepsilon_{11}$ and $\varepsilon_{22}$ linear across $x_3$, it is impossible to meet at the same time the vanishing transverse shear strain condition. These spurious shear stains are sometimes called "parasitic."
As a result the approximated elastic strain energy is wrong, and the finite element patch (under the assumption of complete integration) exhibits a stiffness which is bigger than expected. This point could be easily developed by computing the elastic strain energy of the exact solution, and comparing with the elastic strain energy of a single-element-across-the-thickness patch.
By the way this is a common situation, with any FE approximation. Actually what makes the bending problem pathological is the fact that for a single linear element in the thickness direction $x_3$, the strain energy is dominated by the parasitic transverse shear deformation, so that the displacement components are heavily underestimated.
Remedy for this problem are
structural elements (plate/shell) in which Kirchhoff-Love or Reissner-Mindlin theory is incorporated as an assumed displacement field. (Gradient in $x_3$ direction is exactly modelled by this assumption.)
more linear element in the thickness (more than 5) or use of higher order elements in order to better catch the linear strain/stress gradient along $x_3$, without the introduction of parasitic shear strains.
selective reduced integration, which amounts to integrating the transverse shear field with a single Gauss point at the centroid of the element (where the parasitic shear fields vanishes) while still integrating other terms with more gauss points.
Final notes
Thanks to Bill Green who criticised my first (simplistic) answer and forced me to write a longer (and I hope also better) answer. It was not my intention to give a complete account of mesh locking phenomena, which are indeed well understood and described in the literature.
I'm still convinced that, without going into the full details of shear locking, the starting point for explaining this phenomenon should be the observation that for pure bending the limiting solution for thin plates has very high strain gradients in the thickness directions, compared to the in plane ones. Linear elements are "constantish" in strain. Linear strains for hex elements arise from $x_1\cdot x_3$ (and similar mixed terms): the lack of squared terms (like $x_1^2$) in fact makes it impossible to properly decouple shear and longitudinal strains...
