Why are cooling towers at nuclear power plants shaped the way they are?
A cooling tower has a parabolic shape to improve cooling capabilities. Cooling towers act as a large radiator to cool water used to cool the turbines, much like a radiator in your car.
Inside near the base of the cooling tower there is water distribution system that evenly distributes the hot water from steam driven turbines. As the water is distributed and falls to a pool below it heats the air. As we all know heat rises. The heated air carries a huge amount of water vapor along with it up through the cooling tower. The large opening at the base allows a large volume of air to enter the tower. As the moist heated air rises, the air speed increases due to the constriction made by the parabolic shape of the cooling tower. This is called the Venturi Effect. It act much like the carburetor in your car.
As the moist heated air travels at an increased speed there is a reduction of pressure through the constriction. Above the constriction, the diameter of the cooling tower expand. As the hot moist accelerated air enters the larger space, the moist air rapidly expands. This causes a decrease in temperature and chills the moisture or water content in the air. Much like the evaporator on your air conditioner. The colder water falls down through the cooling tower into the pool below to be reused to cool the turbine. As the heated air exits the top of the cooling tower, it also carries with it a quantity of water vapor caused by the rapid expansion of the air. Thus, when you see a water vapor plume exiting the cooling tower, you can be assured the turbine is in use producing electricity we all depend on for our everyday living.
In other words, a large volume of heated air rises up the cooling tower carrying with it a large quantity of moisture and water vapor. The parabolic shape cause a restriction in the air flow through the cooling tower. This cause an increased air speed. As the air rises above the constriction it rapidly expands and cause evaporation. The evaporation chills the air and cools the moisture and water vapor. The cooler water is heavier and falls back into the pool below to be reused again and again to cool the steam driven turbines.
The rest of the answers here are informative; to get the full picture some reading about the history of the design of these towers is probably helpful.
As others have mentioned, the towers are built this way because they provide a good balance of ease of construction, cooling properties, and tolerance of loads and winds. That is the simple answer. The long answer is: the shapes are the result of many decades of analysis and trial and error, as is a common story in engineering.
This paper by Harte provides an overview of the design and construction of these towers in Germany over the 1990s. This older paper by Krivoshapko was one of the first to do thin-walled physics modelling of these structures. This well-cited paper from 2002 goes into a high level of detail on the design of a 200 meter cooling tower in Niederaussem, going into a lot of depth on the shape optimization. You'll notice that in this case the 'optimal' structure actually isn't really a hyperboloid, it's more like a cylinder on top of a cone. There is no 'diverging' area.
Unfortunately it would be hard to give a simple and correct answer in a few paragraphs; to learn how they are designed how they are, you need to go into a little bit of detail on the design and engineering. But the basic answer of 'it is a good shape for both cooling and structural properties' is probably a fine distillation.
First of all, it's not only nuclear power plants that have cooling towers. Below is a picture of the "Jax Coal Plant" in Jacksonville, Florida:
Why do power plants need cooling towers? Well in the case of a coal plant like the Jax plant, they release a lot of heat burning all that coal. The heat has to have some place to go, hence the plant needs cooling. Same applies for nuclear power plants: they heat up water and that heat needs to go somewhere so it does not build up to dangerous levels inside the plant. Hence nuclear plants have cooling towers too.
Why are the cooling towers that specific shape? Well, they need to be as wide as possible to increase the surface area-- if they were super narrow you'd have a very powerful jet of steam racing through the constriction and that might be dangerous and cause unnecessary pressure on the cooling tower.
Why not just make them wide but straight instead of curved? The hyperboloid shape is more structurally sound. If this doesn't make intuitive sense to you, read the Wikipedia article on Hyperboloid structure or search Google.