Okay, so imagine you have a little toy house made out of Legos. And you have a little toy person that's the same size as the Lego house. Now, if you shine a flashlight onto the house, some of the light will bounce off and come back to your toy person's eyes. This is kind of what happens when scientists shoot X-rays at atoms.
But atoms are much smaller than Legos and X-rays are much smaller than flashlights, so the X-rays don't just bounce off the atoms. Instead, they kind of pass through them, but some of the X-rays get blocked or scattered by the electrons inside the atom. When the X-rays that get through the atom reach a detector on the other side, the scientists can use them to figure out something about the atom they shot the X-rays at.
Now, the shape of the electron cloud around an atom affects how the X-rays will interact with the electrons. This is where the atomic form factor comes in. It's basically a way to describe how much the X-rays are scattered or blocked by the electrons based on the shape of the electron cloud.
Think of it like this: if you had a balloon filled with air, and you tried to poke it with a pencil, the pencil would bounce off because the balloon is kinda squishy and the air inside moves around. But if you had a balloon filled with water, the pencil would go straight through because the water is more solid and doesn't move around as much.
The same kind of thing happens with X-rays and atoms. Depending on the electron cloud shape, the X-rays will either bounce off, pass through, or get scattered in different ways. The atomic form factor takes all of this into account and helps scientists figure out what the electron cloud shape might be, based on the X-ray scattering data they get from an atom.