What is the principle behind the table cloth trick?
The idea is that all objects with mass have inertia. That means the glassware and the dishes all have inertia, and will resist your attempts to change their velocity from zero to a non-zero vector. However, since we live in a physically non-ideal world, the glasses and dishes are acted upon by a frictional force from the table cloth if the table cloth moves.
As you noted in the question details, the force of friction is only dependent on the magnitude of the normal force, not the orthogonal "applied force". So if the magnitude of the normal force is practically constant throughout the trick, the force of friction is relatively constant. The frictional force drags the "crockery" along with the table cloth in the direction of the "applied force".
The degree to which inertia resists movement increases as the size of the change increases. That is, if you try to accelerate a baseball to 120 miles per hour (a very very fast pitch), you will encounter more resistance than if you only accelerate it to 20 miles per hour over the same distance.
The success of the trick is dependent on a balance between the inertial resistance of the "crockery" to moving with the table cloth, and the friction dragging the "crockery" along with the table cloth. The last point about inertial resistance increasing as the acceleration increases is absolutely key in this balance. If you yank the cloth out quickly enough, the inertial resistance is greater than the (relatively) constant frictional force, so the "crockery" stays put on the table and the table cloth slides out. If you pull on the cloth too slowly, inertial resistance is insufficient to overcome the frictional force, and the "crockery" stays on the cloth, falling to the floor.