Brine is a solution of the salt which is usually sodium chloride in water
The sodium and chloride ions actually separate in water, turning solid NaCl into Na+ and Cl− ions that can move freely through the solutions with oppositely charged ions. Electrons are one form of charge carriers and the most common, being that they have a net negative charge and are mobile inside of metals but free ions moving around in a solution also constitutes a current.
When we put two metal poles into a solution i.e. a negative anode and a positive cathode and turn on a battery, you are making a voltage difference between the two rods. As we know from circuits, voltage differences are what drive currents but how this is done is what separates electrons conducting current through a wire and the ionic salt water solution