Shockingly, Wikipedia claims that "The provitamin A beta–carotene from carrots does not actually help people to see in the dark unless they suffer from vitamin A deficiency. This myth was propaganda used by the Royal Air Force during the Second World War to explain why their pilots had improved success during night air battles, but was actually used to disguise advances in radar technology and the use of red lights on instrument panels."
This is only partly true however. One source that Wikipedia cites is this article from the Scientific American (SA), which basically says that the UK's Ministry of Food was happy to promulgate this story in order to persuade people to eat more carrots. They were however well aware that the story wouldn't fool the Germans, who were well aware of the RAF's use of radar and even used it themselves.
But, says the SA: "under certain conditions, eating carrots will help improve eyesight."
To cut a long story short: carrots contain beta–carotene, which is metabolised in the body to produce Vitamin A, and Vitamin A definitely helps to improve your vision in poor light. But this process is not very efficient, and other foods are at least as good at providing Vitamin A – such as "fortified rice, amaranth leaf and goat liver".
The best thing to eat in order to improve your eyesight however is green, leafy vegetables. These help to protect the retina from "high–energy wavelengths of light", and may also protect against age–related macular degeneration – the major cause of blindness in the elderly.
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