To start with, some disambiguation. Solar water heaters do not run off of solar electricity. Ideally, they do not run off any electricity at all. However, they can be the first step to gearing your house towards getting on solar electricity systems. Why?
Well. Again, ideally, they run off little to no electricity.
Solar water heaters convert the solar energy they receive into heat. Many use special polymers and paints to do this, but it's not unlike leaving a cast iron pan of water on the sidewalk on a hot day. In other words, this is a reaction you are familiar with. The use of special materials to form the solar water heater is what makes it more efficient at heating water. There are several different ways it does this, of course, making up several different types of solar water heaters. In the end, most of them are just better suited to one type of weather or environment, not many of them display any clear general superiority over the others. Some are just better if you live somewhere where it gets really cold, some are better if there's a lot more sunlight and it doesn't get so cold- that kind of thing.
One of the first steps to going fully solar is kind of wrangling your household energy, getting what you use down as much as you can. The reason why a solar water heater can be a step towards going solar is that it helps do this. Because most tanked water heaters run throughout the day, as you use your water, having a solar water heater reduces your energy costs by simply... not doing that. Now, solar water heaters do generally have a back up electrical water heater that they use to take over when the water in the tank isn't hot enough. Maybe your household is running through a lot of water one night. So, again, while they ideally use no electricity, many of them will use a small amount.
This still is a great reduction in energy usage. Even if you don't plan on going fully solar (or, perhaps you do, but that is a few years down the road) the reduction in energy costs that getting a solar water heater install now is not to be ignored. Depending on the kind of backup heater your solar heater water uses, you can save between $120 - $240 a year on energy costs, which is not bad, especially if you combine it with other energy-saving measures to reduce your electric bill.
If you want to learn more about solar water heaters, feel free to contact a solar company to learn more.