1- Why is soap made from potassium salts of fatty acids soft?
2- Why does adding a salt solution lead to a precipitation of soap?
I don't have an answer to 1 except to say that's the nature of the beast.
2. Salt causes a "salting out" effect because it is so much more soluble so the salt dissolves and the soap, which was in solution, precipitates. This is a relatively easy and cheap way to get materials that are slightly soluble to ppt. Gotta be careful, though, to make sure the salt doesn't react with what you want to collect.