The myth that carrots can improve your eyesight is simply not true. However, carrots do contain beta-carotene, a substance that your body converts to vitamin A. Vitamin A is the nutrient crucial for maintaining eyesight.
A vitamin A deficiency can cause night blindness and sometimes even complete blindness. In fact, vitamin A deficiency is the leading cause of blindness in Third World countries. But, if you're not already lacking vitamin A in your diet, your vision will not improve no matter how many carrots you eat.
If you eat a lot of carrots, be wary of your skin turning orange. Beta-carotene is a pigment---an orange pigment. It goes without saying: You are what you eat.