Belinda --
"...tallow or lard which wouldn't dissolve in water..." I agree that fat and plain water are insoluble and immiscible, but we're talking about the soap here, not the fat from which the soap is made.
Soap is highly soluble in plain water and does not settle out efficiently. You'd have a fair amount of soap left in solution if you tried to settle it out from unsalted water -- an unnecessary loss.
When salt is added, it turns out many sodium soaps are insoluble in a water-salt solution, if you get the salt content high enough (varies with the type of soap you've made).
That is why KOH soaps are not salted out to concentrate them -- these soaps are too soluble in salty water as well as in plain water so the salting-out process just doesn't work.
Here is a description of the salting-out process from "The Art of Soapmaking" by Alexander Watt, 1884:
"...The oil being now completely neutralised with alkali, the combination in its present state also contains a large quantity of water in the shape of exhausted or spent ley [a very old name for lye]. To remove this, many substances may be employed, but common salt, which answers the purpose admirably, is from its cheapness generally employed. The process of separation, which is generally termed "cutting the pail," is effected by throwing into the pan a concentrated solution of common salt, or a few shovelfuls of the same, each portion being allowed to dissolve before the next is added.
...When sufficient salt has been thrown in, the soap separates from the leys (which also hold glycerine in solution) and coagulates in flakes or granular clots. The soap-boiler, by freely using his shovel — by repeatedly dipping it into the boiling mass and observing its condition — can tell in a moment when enough salt has been added. At this period the ley runs clear off the shovel or trowel, leaving the soap in separated lumps upon its surface. By continuied boiling the clots assume a granular or grain-like appearance, in which condition the soap is said to be "boiled to a curd."
...When the soap has assumed the form of grains or curds, it is known that all the superabundant water — that is, its uncombined mater — is separated from it, and at this stage the fire is drawn or the steam turned off, as the case may be, and the pan is allowed to repose for a few hours to enable the leys to deposit...."