I think it means we're all part of a bigger something, that something being life itself. You are life, and so am I. Hence "I am in you and I am you."
Rumi is saying that in order to grasp that, one must release the ego-mind. The ego is what gives us the illusion of having an individual experience, i.e., feeling separate from others.
I've experienced this in the following way:
My apartment happens to overlook a bus stop in downtown Seattle. On occasion, I've taken a moment to relax and just 'people watch'. I might see a man going to work (and I realize that could be me). I might see a drunk man or a drug addict not having the greatest of mornings (and I can see myself in that person). I might see a woman waiting for a bus with her newborn in a stroller (and I realize that if I were a woman, that could easily be me).
So you see, we are not so separate and different as our ego mind would have us believe.