I imagine that you can interpret this in different ways, but to me, it is saying that what we human beings need most is to feel that we are one with other people -- that we care for others and they care for us.
In this quote, I think "the world" really refers to people. We are miserable because we are separate from others and we can often feel that no one cares about us. At the same time, though, we are capable of making others feel very good through compassion. And what is compassion really? To me, it is understanding how other people are feeling.
So what I get from this is that, when we feel compassion, we are actually becoming more at one with others and we are no longer separate and alone.
Separateness is a state of isolation and disconnection, a state away and aloof from the rest of the world. This state obviously leads to all kinds of misery--anguish and insecurity, selfishness and egotism, diffidence and fear. Separateness is a negative state producing a feeling of vacuity, leading to non-action.
Compassion is. on the other hand, a positive and sympathetic attitude to life. It is a fellow-feeling to be united with all other men and objects of nature. Compassion leads to sharing of the world, showing love and concern for others.
Separateness divides, compassion unites. We all know that 'united we stand, divided we fall'.