Why do fish die in unfiltered water?

Well, that can be answered a few different ways. But the most widely accepted answer would be an explanation of the nitrogen cycle.

When fish excrete urine and feces, it create ammonia in the tank. Yes that's the same ammonia you smell when you don't change the cat's little box for a couple days.

When this waste builds up to a certain point, it literally suffocates your fish to death. What a way to go, laying in a pile of your own excrement...gross.

Filters however provide a distinct advantage to our aquatic friends, and play an important part in breaking down that fish waste.

Filters typically have some sort of medium in them which remove particles from the water...usually foam. But all sorts of media exist for this purpose. When the ammonia begins to build up in the tank, a special type of bacteria actually eat the fish waste and it builds up on your filter foam. These bacteria convert the ammonia into their own form of waster called "nitrites". Nitrites are also deadly to fish.

Nitrites are also harmful to fish. After time goes on though the filter begins to grow an additional set of bacteria, specifically called "nitrobacter" that turn the nitrites...into nitrates. Which are more or less harmless to fish except in very high concentrations. Once nitrates build up and ammonia and nitrites are no longer being produced however, the nitrates can be removed by changing your water.

So now you may be asking yourself, "Well ponds aren't filtered are they?" But the answer to that is also yes and no. Yes because in the wild this process takes place naturally, with nitrates becoming diluted by rainwater and decreasing the concentration. But no the water is not being filtered for particles and chemicals. But we must also take into account that in the wild, fish live in untold millions of gallons of water, sometimes in streams or lakes too. In streams the fish waste is washed away almost instantly though by the steadily moving current, and in ponds and lakes the waste is diluted almost immediately, or the fish can simply swim away from it.

Filters are a very important part of fishkeeping, and allow hobbyists to cheat nature by stealing this important biological process for our own.