This question comes from Section 4 of "Song of Myself," where Whitman talks about the petty concerns of everyday life. The things that are "not the Me myself" are the "trippers and askers" who surround him (those who ask a lot of questions and try to trip him up in his logic), the people he meets, his early life and how it makes him who he is now, all the latest news ("dates, discoveries, inventions, societies, authors old and new"), even such random things as what he eats for dinner and what he wears, who he associates with, whether some woman or man whom he loves loves him back, whether anyone is sick, emotional ups and downs ("depressions or exaltations"), lack of money, battles and wars and attendant horrors, and the anxiety of "doubtful news."
These come to me days and nights and go from me again,
But they are not the Me myself.
These, in other words, are all part of his days and nights, but they don't really matter. They are the little things in life, not worth worrying over. The are not the real "Me." His real Me is the one who stands apart from all this and watches. While he watches, he is "amused, complacent, compassionating, idle, unitary." That is, he doesn't pass judgment. He watches and learns and cares and laughs, but he doesn't pass judgment. Life is what it is. He does not mock or argue: he watches and waits.
Whitman has several selves, and this "Me" is the one who is almost godlike it its patience and distance from the petty concerns of day to day life.