So, we’re all made out of cells, right? And these cells are really super tiny, so why is it when we accidently cut ourselves, like a papercut or from glass, the sharp edge just doesn’t push the cells to the side?
Further, do those edges push aside out individual atoms too, or somehow have we just been avoiding death via splitting atoms?
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Humans are mostly held together fairly tightly (we maintain a fairly rigid shape), even in the soft fleshy bits, so the flesh resists the force being imposed by the piercing object. Some of the cutting does pierce cells, break their resistant cover (membrane), but cells are somewhat squishy (deform rather than rip) so a lot of the work done by the piercing implement is in forcing apart cells rather than ripping them open.
You might get an idea of how the cells “want” to cling together with other cells when you look at a scrape and see partially-detached strips of skin or flesh (flaps of flesh that you can sort of put back into place).
As to the atom-level question, in a certain sense all cleaving is the forcible separation of neighboring atoms that have some sort of attractive force keeping the neighbors together in a group, so the separation is the imposition of a force that exceeds the force of atom to atom attachment, but that view fails to account for the many different ways that atoms glom together, and the very different levels of energy involved between the different atom-atom attachments. Sort of also ignores the different behavior of fluids (liquids, which move around fairly easily) versus solids, and the intermediate plastic-behavior shown by many mixtures or solutions.
As a general idea, it is usually easier to make a clean cut on a low-liquid content item than its “wetter” equivalent because low fluid content means most of the energy is taken at the point of contact rather than being diverted by movement of liquid away from the point of contact. This is why we dry wood before we convert it into lumber or chop it into firewood, for example. Or well-cooked meat is generally easier to cut cleanly than red or rare-cooked meat.
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