In the song "New York City Girl", the singer tells the New York City Girl that "you should know the score by now". Meaning she shouldn't be so naive. She shouldn't think that love is going to be like the Broadway shows, i.e fairy tale like. She's had enough bad experiences to know better, "to know the score by now"!
When a person is generally very astute, and possesses common sense and both self and worldly awareness. For example, someone who is fairly well read, always up to date on current events, can carry on a conversation, and is able to think critically and form his/her own opinions while watching mass media with a critical eye, and is an overall competent individual in life, is someone that knows "the score". People who "know the score" are also characterized by resourcefulness, and the ability to get things done no matter what.