Well, a salty environment makes it difficult for many
microorganisms to survive as water is drawn out of their cells by
osmosis. It may also reduce swelling by drawing out fluids from the
affected area, however there's probably where the antiseptic
properties is helpful.
Salt will also cause the surrounding damaged tissues to turn
hypertonic as the salt dissolves, causing pain and stings. This is
due to one of the neuron receptor types that reacts to electrolyte
concentration changes. On top of that, damaged tissues have more
sensitive pain receptors which means it will not only be more
painful, it may further damage the tissues and render them even
more sensitive to pain.