It suggests a willingness to forgive and forget, and to focus on the present or future rather than dwelling on the past.
Here are some examples of how it might be used:
* "After years of fighting, they finally decided to let bygones be bygones and become friends."
* "It's time to let bygones be bygones and start working together for the common good."
* "We had a disagreement, but I think it's best to let bygones be bygones and move forward."