There is more than one factor that matters in the forces between molecules. The tradeoff between these different factors is complex and can't be reduced to a single simple equation or factor. So some of the factors become more important because of molecular or atomic size counterbalancing the factor of distance between the entities.
The collective force that cause molecules and atoms to attract are often referred to as van der Waal's force (though some sources use a more narrow definition that doesn't include all of the components below). The key non-bonding forces involved are:
- forces between permanent dipoles
- forces between permanent dipoles and induced dipoles
- forces between instantaneously induced dipoles caused by quantum fluctuations
In non-polar molecules it is the third of these that dominates. Crudely stated the force arises because of uneven distribution of the electron cloud in a molecule or atom that happens as a result of quantum fluctuations in the cloud. An uneven distribution of electron density is the equivalent of a dipole moment. These dipole moments create induced dipoles in neighbouring atoms and can interact with other quantum fluctuations in the neighbouring atoms. These create small electrostatic forces between the atoms or molecules.
The reason why larger atoms or molecules seem to have larger forces is two fold. Larger molecules, crudely, have larger electron clouds and more ways to interact (shape matters too and large flat molecules "stick" together more than irregular lumpy ones: compare the melting points of benzene and toluene).
In larger atoms the explanation is simpler. Larger electron clouds are less tightly held by there nuclei than those in smaller atoms. Crudely, larger, floppier clouds can have more scope for the quantum fluctuations than tighter-held smaller clouds (also they have a larger area and that maters to the total force). So the size effect increases the strength of the force rather than reducing it as might be expected from simpler arguments.
I'm aware that this is a slightly superficial explanation, but a better one would probably involve enough quantum mechanics to make the average reader's head explode. I think i've captured the key intuitions that matter.