Since the solution is obtained using the stiffness method of analysis, if you look at the basis of calculation of the node displacements, it will be apparent that it is affacted by the subgrade modulus as well as the thickness of the elements representing the mat.
The node displacement is obtained by the well known stiffness equation
[K]*{d} = {P}
which leads to
{d} = [Kinv]*{P}
where
{d} = matrix of node displacements
[K] = global stiffness matrix
[Kinv] = inverted global stiffness matrix
{P} = matrix of equivalent nodal loads
The global stiffness matrix [K] is assembled from two terms - A and B
A = stiffnesses of all line members, plate elements and solid elements in the model
B = stiffness of all the springs specified at the supports. It happens to be soil springs in this case.
The thickness of the elements affects A
The subgrade modulus affects B
Keeping the load unchanged, as you increase the thickness of the mat or the subgrade modulus, A and or B increase, and thus, the overall system becomes more and more stiff. The deflections will become more uniform, the maximum base pressure will become less since a load is distributed over a larger number of springs (a wider area of the soil).
The PLATE MAT shown in that example is very rudimentary. All elements are exactly the same, and the load is a uniform pressure on all the elements. Consequently, every joint has to displace by the exact same amount. The hand calculation is taking advantage of these facts. Had the mesh and/or the loads been more irregular, the author wouldn't be able to calculate the displacements or pressures using the approach he/she has used.