The short answer to this question is not really.
Carrots are a good source of vitamin A, a deficiency of which can lead to night blindness, which causes the eye to adapt very slowly to changes in light.
The retina of the eye is made up of light sensitive cells called rods and cones. The cones pick up detail and color, but need plenty of light to function correctly. The rods however, cannot distinguish color at all but need less light, and so are used for night vision. Rods contain rhodopsin, a light sensitive chemical, the key ingredient of which is vitamin A.
The easiest way to treat night blindness is to increase your intake of vitamin A, most commonly found in carotene. Carrots contain carotene, but apricots and dark leaved vegetables such as spinach and bilberries contain higher levels of carotene.
However improving defective night vision is very different than making normal night vision better. And so eating lots of carrots will not help you see any better in the dark.
During the Second World War, Group Captain John Cunningham (1917-2002) got the nickname 'Cats Eyes Cunningham'. His 604 squadron operated at night, and the British Government encouraged rumors that he was able to see in the dark because he ate so many carrots. Although this disinformation was deliberately designed to cover up the fact that he was testing the top secret and newly developed airborne radar system.
It seems unlikely that the Germans believed this rumor; however it did encourage a generation of English children to eat the one vegetable that was in constant supply throughout the war.
However the government started to overdo the carrot propaganda and carrots became 'these bright treasures dug from the good British earth'. A 1941 recipe for carrot flan, along with carrot jam and marmalade, failed to find their way onto the British breakfast table. Although carrot jam is popular in Portugal.
And so, carrots do not help you see in the dark unless you suffer from defective night vision, in which case the vitamin A found in carrots can help improve your condition. However you would be better off eating apricots, spinach or bilberries which contain higher levels of vitamin A.
So next time someone tries to get you to eat carrots by saying 'they'll make you see in the dark', you will know better. Though carrots, like most vegetables, are very good for you and so you should really eat them anyway!