Sorry for organized the question badly. My supervisor forced me to read chapter $3\frac 12$ of the reputed book Metric structures for riemannian and non-riemannian spaces written by Mikhail Gromov, which is much more like a review about his mysterious thought about mm space, $\square_\lambda$ metric, and much more example with a background about the subject, such as geometry of groups, which I never learned before.
After I tried to understand some basic context about metric geometry, such as A course in metric geometry by Burago, I still could do nothing about the chapter $3\frac 12$. So I sincerely ask you there for some advice. If there is some text that could be easily understand about those mysterious? Or what should I do for getting some advantage in studying those context?
P.S. For someone who has the same question, at least I found a lecture note about $3\frac 12$. Hope it would help.