It means, (1) to start a conversation with someone you have not met before or (2) to make people who have not met before feel more relaxed with each other. And also, (3) to attempt to become friends with someone.
- I never know how to break the ice with someone I have just met at a party.
- We played a couple of party games to break the ices.
- He tried to break the ice, but she was a little cold.
- A nice smile does a lot to break the ice.