Cells contain less salt than sea water, and cell membranes are semi-permeable. That leads to osmosis drawing water out of the cells, drying out the fish.
Sea-dwelling creatures have different ways to deal with this. Most fish can "pump" salt from their blood into the water in their gills, in addition to their kidneys filtering salt.
The other strategy is to eat fish that have already done all that work, after all most animals are mostly water of compatible salt content.
(Freshwater fish have the opposite problem, without active salt management they would explode from osmosis drawing water into their cells).
Basically the same reason it is for humans – it puts a lot of strain on their kidneys. I believe that cetaceans are thought to be able to drink seawater as a last resort, but get most of their water from eating fish. The fish don't have the same salt content as the water that they swim in.
Why is saltwater such a hypothetical problem for them? How are ocean-going fish and shrimp "fresh water"? Those are going to be saltwater too.