This article is a nice overview but makes it more complex than it really needs to be. If you know a little bit of science, you could figure out relatively easy why it would have some beneficial effects. I'll make it easy to understand:

Osmotic pull: remember, "where salt goes, water flows". Take a cell (any kind of cell) there is an amount of water and "salt" ions inside the cell, just like there is outside of the cell (interstitial space). Our bodies are constantly in balance to maintain a balance within/outside the cell. If an area outside of a cell has a higher concentration of salt ions, it will pull water across the cell membrane because h20 is attracted to Na++(salt ion), causing the cell to shrink. Visa versa if the concentration of Na++ is higher in the cell, it pulls water into the cell causing it to swell.

Apply osmotic pull to inflammation. Why does the doctor tell you to gargle/swish with salt water when you have a toothache or swore throat? Because you have inflammation: inflammation is the buildup of fluids that have transported other proteins to the site of irritation to help heal. Essentially gargling with salt water creates a higher concentration of salt/ions to create an osmotic pull and relieve the inflammation site by pulling the inflammatory fluids out.

Apply this knowledge to epsom baths, sure skin is 'waterproof' but how do you think you sweat? Skin pores! Epsom baths just help pull out inflammatory fluids(toxins) out of your body. When you bathe in a hot tub, your pores open up allowing you to sweat. Ever tasted your own sweat when working out because it drips in your face? It is salty, right? Because salt is coming out of your pores and pulling water with it. Relaxing bc you dont have as much swelling after epsom bath.

Hope this helps a bit.