Electrons are the currency of exchange between atoms essentially, except when they make a transaction they're "bound" (not by a contract even though I want to say that to continue the metaphor lol) to one another.
The giving or receiving of an electron can cause a large release of energy so when Na is on its own it has the capacity to make a transaction that will result in some release of energy, if the transaction is already completed (as in salt) then the capacity to react is reduced greatly as the available electrons are now being "used."
In the case of Cl, as with other elements like Arsenic, they can react with molecules in your body that really needed a different atom to fit, but Cl is also able to take that space. Since it wasn't the atom that was needed for that bodily function it can cause issues (death) as it won't be working "as intended."
I'm not an expert, just an idiot factory worker, but pretty much all reactions are the interaction of electrons between atoms.
HCl (Hydrochloric Acid - Very dangerous)
NaOH (Caustic Soda - Also very dangerous)
Combine to literal Saltwater (H2O and NaCl).
Interestingly, because the reaction goes both ways, in any amount of salt water, there will be a miniscule amount of Hydrochloric Acid and Caustic Soda. The concentration is so low because they immediately react again, going back to water and salt.
It's the OH- and H+ (or H3O+) ions that make the acid and base react the way we think about them. You don't need salt for those to form spontaneously in water. Pure water will do it too. But there are many molecules in even a small volume of water, and the average behavior at pH 7 is such that you won't get acid or alkaline burns.