What does it mean when a guy tells you you deserve better?

It means that your partner feels they arent good enough for you. do they show any signs of insecurity. maybe you should talk or do something together. Does your partner feel guilty perhaps for something they have done, and feels you deserve better? I suggest you bear these things in mind and have a conversation, maybe cook a meal and chat. But dont make any rash decisions! It could just be they cant believe they have someone like you who loves them! :)