For the sake of understanding, try to drop any preconceived metaphysical or theoretical ideas of how reality is. Focus purely on your direct sensuous awareness of reality, your conscious experience.
"I know you." Consider, in the context of your direct conscious experience, the meaning of that sentence. What you are really saying is "I know what I think of you, my thought of you, my image of you." So in terms of sheer intuitive understanding, who you know, or perhaps what you know, is an expression of yourself--you know your own interpretation/compartmentalization of your conscious experience.
How could I, the image that exists in your mind, possibly exist without you? I am a projection of your mind, a projection of you. So who are you? You're essentially who everyone tells you who you are. And the circle continues by realizing that everyone who told you who you are is simply a projection of your own consciousness, your own self--this is the ultimate self that Hinduism boils down to, and this is the relationship between all beings.
So when you go ahead and say "I know you are separate from me." What you are saying is "I know my idea of my own conscious experience is separate from me." So you truly cannot love yourself, and likewise know true peace, unless you love everything as yourself because every "thing" is just another image you've created of your own conscious being. You've created everything and are everything, eternally."