Great question. At first it doesn’t make sense. Air has a lot more oxygen in it than water does.

A couple of key concepts are involved here. One is surface area. A single cell can get all the oxygen it needs just by having oxygen diffuse in through the membrane over its surface. Larger things, like fish, have to collect all the oxygen in one area (gills for the fish) and transport it to every cell (in blood for the fish). The more surface area there is in the gills, the more oxygen can be absorbed. Gills are made of many thin, wrinkles and finger-like or hair-like projections to increase surface area.

The next concept is density. Things that are less dense float on things that are more dense. Water is more dense than air, so many things that sink in air can still float in water.

Now we’ll put these together. In water, the projections on the gills float, so each is surrounded with water from which it can absorb oxygen. On land, these all collapse together, blocking most of the surface area. So they don’t work well and the fish suffocates.

Lungs increase surface area by having many tiny pockets, which don’t collapse in air.

If these kinds of questions interest you, you may want to explore physiology or fish biology.

This actually has to do with one of the common "themes" in biology: the relationship of structure and function. The short answer is that gills collapse. They need to have a large surface exposed to water to collect enough oxygen to transfer to the blood. When out of water, the gills crumple and stick together as they dry--think of wetting some tissue paper and letting it dry. This makes all the surfaces stick together, and reduces how much oxygen can be transferred in.

I'm going to add a bit more to this because it's one of the really, really cool things that you see everywhere once you notice it. A main "driving force" in biology is the transport of the chemicals necessary for living. Tiny cells manage this more easily because oxygen and other necessary chemicals can move (or "diffuse") through them easily, like food coloring through water. Humans are much larger, and have to have complex systems like the heart and the circulatory system to make sure that the oxygen we need outside of us makes it inside.

In biology there is always a competition for the most efficiency. Look at plant leaves, for instance. A tiny leaf will not absorb as much sunlight (smaller surface), but too large of a leaf will require a huge stem or trunk to hold it up, and the sun will make it so hot that the water will evaporate out of it. So it is the opposite forces of cost (to build the structure) and benefit (receiving the energy it provides) that dictate if a leaf will be built. Scientists have made models that use math to determine the "cost" of building a leaf vs. the "benefit" of the sunlight it receives, and have been able to duplicate and predict exact leaf shapes based on their math! It's really cool, and if you like math it's even cooler.

To wrap this up, you can look anywhere in nature and see where the effects of cost and benefit of surface area changes happen. This is why the brain has bumps all over it actually: to maximize the surface area of the brain for the most neuron connections, but at the same time to keep the brain itself fairly small and lightweight. It's why mitochondria have the crumpled inner membrane (maximum surface in a small space). It happens on every scale. Ever since I learned this I see it literally everywhere I go.

So basically, gills function by trying to have a large surface in a small space, and drying them out sticks the surfaces together and ruins it.

Many species of fish use a system of gills to obtain their oxygen from the water. Gills are designed with a high surface area of capillaries that bring the blood close to the water to exchange oxygen and carbon dioxide. As a consequence of this, gills are extremely thin and very delicate, like having thousands of pieces of tissue paper very close to each other. These gills operate by in-taking the water through the mouth and having water flow through them at a steady rate to keep the gills separated and functioning, but when the gills are taken out of the water, they effectively collapse on each other, much like how wet tissues stick to each other. As a result, a fish out of water cannot absorb oxygen and will quickly asphyxiate. Some species of amphibious fish do have modified gills and are capable of surviving on land for up to a few days, the mudskipper being one of them.

You're right, fish "breathe" by aerobic respiration, but the way they use oxygen is different than how we use oxygen. We inhale air, use the oxygen and breathe out carbon dioxide. Fish use gills to "breathe" by dissolving the oxygen from the water into their bodies. Also, air is way more concentrated with oxygen than water is, so water is also better for fish in that way. Fish need the water so they can dissolve the oxygen to "breathe", whereas humans need air (and the high oxygen concentration of the air) to inhale their oxygen content.

Fish use gills to breath, and gills can only bring in oxygen when moist, because the chemistry that allows blood to carry oxygen happens in water. Out of water, a fish's gills dry out, and then the fish asphyxiates.