When you dissect a frog in science class, you cut it open to look at its inner organs. To dissect is to break something down to look at its parts.
Dissecting something allows you to look at it closely and understand it better. If your soccer team loses a match, you might want to dissect the game afterward to try to find exactly how and when things went wrong. Television analysts will often dissect a political speech to understand what was really being said, how true the statements were, and how well it was done.