What does the saying or quote "i am my brother's keeper" mean?

The dictionary of cultural literacy offers:

"Cain's words have come to symbolize people's unwillingness to accept responsibility for the welfare of their fellows — their “brothers” in the extended sense of the term. The tradition of Judaism and Christianity is that people do have this responsibility."
(http://www.answers.com/topic/am-i-my-brother-s-keeper)

To my mind the problem is not the definition, it's the consequences. If "keeper" means that we have a duty to consider the welfare and happiness of others when acting ourselves then yes - that is fine. If it is interpreted as justifying taking some sort of paternal control over others then that is not fine.

One of the hardest lessons to learn is not how to be responsible for others but how to trust them to be responsible for themselves.

Actually the quote is from Genesis, the first book of the Bible. Cain had killed his brother, Abel. When God asked Cain where his brother was, Cain replied: 'Am I my brother's keeper?'

Probably something like,"Why, am I supposed to be babysitting him am I?"

One who takes care of another person or a sibling.

someone who's "got your back"

It means that you take care of them.
not familiar with the story, but I've heard the phrase before, and have gathered as much.

i'd say protector