The quotation should read:
The best laid plans of mice and men often go awry.
This was written by a guy named Robert Burns in 1785, not John Steinbeck. It is from a poem I have attached below called To a Mouse. The verse the line is from talks about how we often make great plans or schemes hoping that they result in joy, but they actually result in pain.
So to take a look at how this compares to Steinbeck's Of Mice and Men, two of the main characters have a dream. The dream is never achieved throughout the story and both endure an extreme amount of pain.