How does Conan know that something is cut recently? In many episodes, Conan sees a tree branch or a rope and after a quick look, he states that it is recently cut which means it is done by the culprit.

How does he know that?
A cut rope or branch (or any piece of wood) will start to discolor almost immediately.  If you know what a fresh cut and non-recent cut look like it is not difficult to spot anything in between.