Remember that like charges repel each other. That is, positive repels positive and negative repels negative. Similar to how the north poles of magnets repel each other and south poles repel. However, at the atomic scale, protons, which have positive charge, are more influenced by the "Strong Force," which binds them close together. If they were to be separated ever so slightly, then the electromagnetic force would take over and they would repel each other like you'd expect.
Neutrons are also held together via the Strong Force, but don't have a charge so when separated, don't have an electromagnetic force pushing them away from each other.
However, electrons act differently. There is no "Strong Force" just the electromagnetic force. So, they keep a great distance from each other.
So in an atom, protons and neutrons stay close to each other, taking up little volume, while electrons take up a lot of volume.
BTW, the reason why electrons and protons act differently when they are close together is because protons are made up of smaller particles the carry this Strong Force. For electrons, there is no smaller constituent. And therefore, all you have is the electromagnetic force to influence it. That's it.
