To say something succinctly is to say it briefly, fitting your points into one concise statement. It's a talent rarely enjoyed and much appreciated. You succinctly stated your case and sat down to great applause, not to mention great relief.
The adverb succinctly derives from the Latin succinctus,"to gird from below, or gather up one's clothes." It referred to gathering together loose clothing and cinching it with a belt. Speaking succinctly reduces the volume of a speech to the main idea, like a belt might reduce the visual volume of an outfit. If your audience seems to be drowning in a pile of verbal fabric, you need to speak more succinctly.