we are claiming that the table is exerting a force towards the object. Yeah sure! What kind of atomic interpretation could this have?
There is no "interpretation" here. The force you are looking for is one of the 4 fundamental forces of the universe: the electromagnetic force, and you are quite familiar with it.
This force governs almost all everyday physical interactions except gravity-based ones. This means all possible effects, phenomena, movements etc. including keeping atoms, molecules and whole bodies together, all motion that's not caused by gravity, everything going on in liquids, gases etc., everything having to do with pressure. It is based on charges which repel or attract each other.
In your example, if you put an apple over a table, gravity pulls it down towards the table. As soon as the atoms which make up the apple get close enough to the atoms of the table, repulsion (of the electron hulls of the atoms in the separate objects) and attraction (of the atoms/molecules within the same object) lead to the apple basically floating over the table, in some sense. The gory details are complicated, but if it interest you, you can go down a rabbit hole of intermolecular forces or intramolecular forces. These are all non-fundamental, secondary forces "created" by electromagnetic effects.
In fact, depending on which physicist you ask, objects never touch at all (see this video for a very entertaining demonstration), but all discrete objects float next to each other, attracted and repelled by a combination gravity and electromagnetism.
So in this case, the force of gravity pulls down, the force created by electromagnetism pushes up, and when they are equal, the net overall force equals zero, at which point Newton's First Law kicks in and makes sure that the apple stays where it is.
This aspect of the electromagnetic force is a bit hard do "feel" for us, but all other motions you see in everyday life are based on it too. Some easily accessible are for example magnets, or statically charged hair which stands up (or even follows your hand if you move it over their head). Obviously these seem quite different from a force which keeps objects from penetrating each other, but at the end of the day it's the same. Charged objects creating electromagnetic fields, attracting and repelling.
As mentioned in the comments, there is of course a lot of other stuff going on, e.g. the Pauli Exclusion Principle, but I would like to focus on the electromagnetic force as I find it quite fascinating how a lot of things we see can explained with it alone - from the atomar level to galaxies spewing out streams of matters...