Very fresh fish does not, in fact, smell fishy – which is handy knowledge when dining on sushi. If it smells, send it back!
As a fish decomposes, nitrogen-containing chemicals called amines are released. Some of these are quite volatile and extremely smelly. This chemical release, and thus the smell, intensifies the older the fish gets.
An excellent reason to gut a fish quickly is that fish guts have chemicals that help break down fish, since many fish eat other fish. The contents of fish guts can speed up decomposition and the associated release of smelly amines.