Flow is important for two big reasons:
1. It removes waste from around corals and brings in new nutrients. Unlike mobile animals that can move to cleaner conditions with more food, the corals rely on the current to do this. SPS corals which grow in tight formations need more flow to bring food to all the polyps and to remove the wastes. Corals that rely more on filter feeding (such as softies) prefer the flow to be a bit gentler because the need time to filter out nutrients. SPS corals uses stinging cells to capture microscopic organisms, so they want as many of these organisms brought to them as possible. LPS coral are in the middle of the two, they also capture organism by the stinging cells but high flow will damage their tissues.
2. Gas exchange. Liquids dont hold a large amount of gasses, and seawater with all the other dissolved minerals and organics holds an even smaller amount of gas when compared with freshwater. So every molecule of CO2 produced by fish when they respire (yes, fish do respire. Respire doesnt mean to breath in and out, it means to break down sugars into CO2 and water) takes the place of the oxygen molecule that could be in the water. Now, desire for the world to be a equilibrium means that the CO2 molecules in the water want to leave because the concentration of CO2 in the water is higher than that of the surrounding air. Oxygen also wants to diffuse into the water because the concentration of oxygen in the water is less than that of the surrounding air. Which is good for us, because otherwise our fish and corals would just suffocate to death. Bad news is that it happens really slowly because a concentration gradient is developed in the water and at the gas/liquid interface. So the molecules dont "see" a big difference in the concentrations around them and there is little "motivation" for them to diffuse in/out of the water.
So where does flow come into this? Flow prevents this concentration gradient from developing, so the molecules have a greater driving force to move in/out of the water. The more flow, the more you can disrupt this concentration gradient and the more oxygen you can keep in the water and the more you will keep the CO2 out. More oxygen means happier fish and corals. Also having a powerhead pointing at the surface, making the surface ripple, will help move molecules. It basically makes it easier for them to move in and out.
You can test this for yourself. If you turn off the flow in your tank for a few days, you will see all the fish gathering at the very top of the aquarium because that is where the high concentration of oxygenated water is.