The reason is not to save fuel.
There are a couple of primary reasons for requiring window shades to be closed during flight:
People Need to Sleep
Nearly all East-West long-haul flights (i.e. those crossing lots of time zones) will require window shades to be closed and will turn off cabin lighting for most of the duration of the flight (usually for the entire time between meal services.) This is because almost every passenger wants to sleep and many of them need to sleep on different schedules.
People on short business trips often want to remain on or close to their home time zone in order to reduce the effects of jet lag. People on longer trips usually want to adjust to their destination's time. As a result, these people will need to sleep at different times, so the lights are kept off - and window shades kept closed - for the vast majority of the flight in order to accommodate whatever sleep needs people may have.
An eye mask can certainly help, but some people don't like wearing them and also people need to be able to get up briefly to use the lav or grab a brief snack and then want to doze off back to sleep. This will be much easier if the cabin is kept dark.
Bright Light / Heat
Even on short flights, at certain times of day and when traveling certain directions, a lot of sunlight can enter the cabin. This can make the cabin uncomfortably hot, especially for people near the windows and/or in direct sunlight. Additionally, the light can be very bright if it's shining in directly through the windows. Even on short-to-medium length daytime flights, some passengers will want to nap, but even those who aren't napping generally don't want the sun shining directly into their eyes. Note that, while the sun from your window may not be shining directly onto your face, it very well could be shining directly into the face of other passengers on or near your row, even across the aisle, depending on time of day (and year) and flight direction.
There's actually enough UV to cause sunburns on directly exposed skin, since airliners fly above the majority of Earth's atmosphere (the majority of the gas in the atmosphere, that is, not a majority of the height of the atmosphere.) There's not nearly as much atmospheric absorption or reflection of the sun's light at 30,000-40,000 ft as there is at the surface, so direct exposure for extended periods, in addition to being annoying, can be harmful.
While this reason isn't as important as the other two, a third more minor reason for closing the shades is so that people can see their IFE screens. The screens can be difficult to see if there's sunlight shining on them.