Yes, freshwater dips do kill many parasites within a couple of minutes. This works because they cannot stand the osmotic pressure they endure in FW. If a human drinks 8 oz of saltwater, he must then consume at least 16 oz of freshwater to replace it, because the SW actually dehydrates him. Same principle for the parasites, but they fall off the fish and die within a few minutes.
Fish are able to hold up for a minute or two, but after that they can suffer consequences. White spot and black spot in particular are treatable with FW dips. Kit is right, though, that any free swimming parasites in your tank can (and will) reinfest the fish.
A lot of LFSs recommend FW dips because they know that their tanks have been exposed to Ich, Velvet, and other parasites in the past, and they can't gaurantee that the fish you are buying hasn't been exposed.