What the salt does is change the osmotic pressure within an organism. If you think back to your school science classes, maybe you'll remember the idea of diffusion - that molecules want to be spread out evenly. This is true, even when there's a membrane (cell membrane in this case) in between. Water can move freely to either side, but salt cannot. So for the concentration of salt in the water to be the same on both sides of the membrane, it's the water that has to move.
In freshwater, there's not a lot of salt in the water. All organisms need some salt to be healthy (it's used to produce stomach acids, to conduct nerve impulses, and to move other minerals in and out of cells). Fish have specialized cells on their gills specifically from removing small amounts of salt from the water, and they're capable of storing this inside of their cells. Parasites and fungi (what salt is generally used to treat) don't have the same mechanisms. So when they encounter salt, the water is pulled out of their cells to try and make the salt content inside and out balance. What literally happens is that these organisms dehydrate in a tank full of water because the salt pulls the water out of their cells.
In saltwater, you can do the same for parasites by lowering the salt concentration so they fill with water and burst.
Another benefit of salt in a new tank is that it eases stress from nitrite exposure by binding to the sites where nitrite is absorbed - nitrite interferes with the uptake of oxygen, so when nitrites are high, fish will be at the surface looking like they're gasping for air.
For most fish, a tablespoon per 5 gallons is enough. Some species (especially livebearers) can tolerate much higher levels, while others (tetras, scaleless fish, catfish) are more sensitive, but all can tolerate the amount listed above.
That said, if your softener is raising the salt level in your drinking water more than a minuscule and measurable amount, there is something wrong with the softener, sodium is rinsed to drain, as it is the chlorides that is desired in sodium chloride that is used for the exchange process.
Ive filled and refilled my freshwater tanks with softened water for ever. RO water is actually bad to use since untreated the PH level is lowered when processed through the membrane.
Yesterday while at my favorite fish store I couldnt help noticing the packages of freshwater tank salt for sale and a dispensor in 1 or 2 tanks.