The short answer is that water evaporates in the sun, as it does in any hot place, because the heating of water causes the water molecules to move faster and faster until they break free from the bonds that hold them in place as a liquid. The molecules then escape from the liquid into the surrounding medium as free gas molecules.

A water molecule, or H₂O, consists of two hydrogen atoms and one oxygen atom. The hydrogen atoms tend to have a slight positive charge, while the oxygen atoms have a slight negative charge. As a result, the hydrogen atoms in a water molecule attract the oxygen atoms in different water molecules. When these attractive forces are strong relative to the movement of the water molecules, they form bonds between water molecules called hydrogen bonds. When the water molecules are moving slowly, these bonds between water molecules hold water together as either a solid (ice) or a liquid. When the water molecules are moving fast enough to overcome and break these bonds, water becomes a gas.

Therefore, water can occur typically in three phases: solid ice), liquid, and gas. When the molecules are moving very slowly, they are typically arranged by the hydrogen bonds into a crystalline ice structure, the solid. When the molecules are moving slightly faster, breaking some but not all of the hydrogen bonds, water is a liquid and flows freely but the molecules are still loosely connected to each other. When the molecules are moving even faster, virtually all of the hydrogen bonds may break, causing the water molecules to free themselves from each other and become a gas (i.e., to evaporate).

There are two ways in which you can cause water molecules to move faster in order to break these bonds: you can heat up the water, or you can decrease the atmospheric pressure on the water. By heating up the water, you add energy to the water molecules and cause them to move faster (temperature is just a measurement of average molecular motion). When the molecules are moving fast enough to break the hydrogen bonds, the water can evaporate, turning from liquid to a gas. This is what happens when you boil water on a stove.

By decreasing the pressure on the water, you are essentially decreasing the speed that the water molecules have to move in order to break their hydrogen bonds, and therefore you are decreasing the temperature at which the water evaporates.

When you boil water on top of a high mountain (at a high altitude), you can cause the water to boil and evaporate (to change from liquid to gas) at a much lower temperature than when you boil water at sea level. This is because the atmospheric pressure is much lower on top of a mountain than at sea level, and the water molecules do not need to move as fast in order to break their hydrogen bonds.

On the sun, I do not know what the atmospheric pressure is, but the temperature is VERY hot indeed and is certainly the reason that water evaporates here. In fact, water molecules on the sun may move so fast that they not only break their hydrogen bonds and become gas, but they may also break apart into their constituent atoms: hydrogen and oxygen. If water molecules break apart on the sun into hydrogen and oxygen atoms (called fission), they would no longer have any of the properties of water. They would probably become part of a soup of neutrons and electrons called plasma; it mostly contains neutrons from hydrogen and helium atoms.

When sunlight shines on water it transfers energy to the water in the form of heat. When water heats up the oxygen and hydrogen molecules gain this energy and start moving faster. Once the energy is high enough the water molecules will break apart and change from a liquid to a gas state causing evaporation.

Water exists in a state of equilibrium with the air, in which water in the air continually condenses into water and in which water continually evaporates in the air. If more water is condensing than evaporating, then the water body will swell, and if more water evaporates than condenses, the water body will dry up.

Increasing the temperature (e.g. adding sunlight) increases the rate at which water evaporates into air and decreases the rate at which water in air condenses back into liquid. Thus, if the temperature is raised to the point where the air can now hold all of the water in a vapor state, the body of water will eventually disappear (and the higher the temperature, the more rapidly it disappears).