To keep your eyes peeled is to remain alert or watchful. It means the same as keep an eye out for something.
- Keep your eyes peeled while crossing the road.
- Keep your eyes peeled for stray dogs. They can be quite dangerous. = Keep an eye out for stray dogs.
- Keep your eyes peeled while taking the subway. If you don’t you could get robbed.
- Keep an eye out for pickpockets while taking the subway.
- While you are on the road on a Saturday night, keep your eyes peeled for drunk drivers.
- While you are on the road, keep an eye out for drunk drivers.
This expression seems to have been in use since the 1820s. According to historians it was Sir Robert Peel who established the first organized police force in Britain. They were called peelers and their duty was to keep their eyes peeled. In other words they had to remain observant.