The expression “to keep one’s eyes peeled” means to be very observant or extremely alert; to keep a sharp lookout.
No record of this American expression has been turned up earlier than 1833, but the fact that it then appeared in a newspaper (The Political Examiner, of Shelbyville, Kentucky) is a fair indication that it was already well known to any reader.
The meaning was not explained.
The passage read: “I wish I may be shot if I don’t think you had better keep your eyes skinned so that you can look powerful sharp, lest we get rowed up the river this heat.”