In the atom, electrons zip around the nucleus. The electrons always travel at the same speed. They can change direction, they can spin, they can even move to a neighboring atom. But they always travel at the same speed.
But they're actually not travelling at the same speed. We know that time slows as we approach a gravitational object. So the electron that is zipping around the nucleus is actually slowing down near the gravitational source.
All of the electrons in the atom would be affected in the same way. As the electrons go around the nucleus, time slows as they near the gravitational source. The electrons think they're going the same speed, but the time differential makes that impossible.
Because the electrons spend slightly more time on one side, that side has a partial negative charge. More electrons would mean a stronger pull on the nucleus. It is this pull that is the actual source of what we perceive as weight.
Take a pencil and put a dot on a piece of paper. How many atoms are in that dot?
If you were able to stack all of the atoms, one on top of another, to make one long string, that string would be long enough to be wrapped around the Earth millions of times.
And that's just a dot from a pencil. How much does that dot weigh? Anything with measurable mass has many trillions and trillions more atoms. (Actually, the number of atoms in one gram of pencil lead is over 600000000000000000000000. That's 23 zeros.)
So the individual pull of one atom is infinitesimal. But together, with all the other atoms, it is not insignificant.