Interdependence was popularised by Steven R. Covey in his "Seven Habits ..." book [1]. Basically, it means that human beings (as well as organisations, and nations) need each other to thrive.
From You to I to We.
First is dependence - the ‘you’ phase, where the child looks to others for its safety, wellbeing, and pleasure. If someone withholds, the child is deprived. With caring others, the child thrives.
Second is independence - the ‘I’ phase, where the child develops into adolescent and young adulthood. If all goes well, the young adult develops independent traits - taking responsibility, making sound decisions, and becoming self-reliant.
Third is interdependence - the ‘we’ phase, where the independent adult chooses to increase their circle of concern, to include ever widening groups of people.
Interdependence is the recognition that people – and the social system within which they exist – are mutually dependent. We cannot function alone, we need social cohesion and the work of others to thrive.
And for the group or social system to be viable it needs us to make our contribution. Only independent people can choose to be interdependent.
While this staging is neat, real life is messier. Children may be more independent as toddlers than as teenagers, and teenagers more interested in looking after their peers and the world at large than adults. But, in general, yes, adults are expected to care about others and themselves - it is a sign of greater emotional maturity.
Compare with Vidura Niti in the Indian/Mahabharata context.
NOTES
[1] Covey, Stephen R. 2004. The 7 Habits of Highly Effective People : Restoring the Character Ethic [Rev. ed.]. New York: Free Press.