Water hardness isn't affected by filters. It's the presence of metal ions and is removed by converting them to salts. Hard water can taste good, and is perfectly healthy to drink, but it prevents soap from lathering and in areas with lots of moving water, it leaves mineral deposits, like stalactites and stalagmites in a cave. Usually it's treated to make cleaning easier and to prevent deposits from building up.
If you have to add salt to your dishwasher, it is to treat hard water, and in areas where it is needed, if you don't add salt the dishes will build up a cloudy mineral layer that is especially visible on glass and flatware as a matte finish when dry.
Seconded, although some still go for under-the-counter 6 stage water filtration systems with a small tank for drinking water. There are also 4 stage filters for the water as soon as it enters the house (very basic ones that do not restrict the waterflow as much compared to the drinking water filters) and 2 stage under sink solutions you can use on a per-sink base. I have seen people use all kinds but rarely.
If you have to add salt to your dishwasher, it is to treat hard water, and in areas where it is needed, if you don't add salt the dishes will build up a cloudy mineral layer that is especially visible on glass and flatware as a matte finish when dry.