Succinct is the past participle of the Latin succingere, which comes from sub + cingere "to gird." So the verb means "to gird up" and the past participle means "girded up." The sense extended to mean "confined by a girdle" and then "compressed into a small space" and eventually "verbally concise or terse." Weird.
And I originally thought that I didn't recognize the root cingere, but I think I do—the French word for "belt" is ceinture, which presumably comes from the past participle, cinct.
Edit: Ooh! And the word cinch is related, too. That comes via the Spanish cincha "girdle," which comes from the noun form of cingere, cingulum. The meaning of "an easy thing" comes from the sense of "a sure hold." The verb comes straight from the belt/girdle sense of cinch.