Be alert.
Some think that keep your eyes peeled is a reference to certain fruits or vegetables that improve if peeled - e.g. oranges. Perhaps more likely the reference is not so much to peeling back fruit as figuratively keeping alert by peeling back the eye lids so that vision is unimpeded. This seems a likely option given that an alternate of the expression is to keep your eyes skinned.
Another possibility (or perhaps just a related pun) is that the expression comes from the early days of the British police. In 1829, Home Secretary Sir Robert Peel, established the Metropolitan Police Force in London, the first professional police force. Before long the policemen had picked up the nicknames 'Peelers' and 'Bobbies' for the unpopular fact that they reported directly to Peel at the Home Office. Bobbies lives on to this day as a friendly nickname for the police in England. Peelers is not used but may have left its marks in the expression to keep your eyes peeled, as per their founder's instructions.
Although probably not the first use, in 1853, The Daily Morning Herald in St Louis wrote 'Young man! Keep your eye peeled when you are after the women'.
More expressions and their source
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