The word means "of Laconia," and the region is the whole story.
Etymonline dates "laconic" in the sense "concise, abrupt" to the 1580s, literally "of or pertaining to the region around ancient Sparta." It came through Latin Laconicus from the Greek Lakonikos, which comes from Lakon, a person from Laconia.
The plain geographic adjectives are slightly older. Etymonline records laconical and Laconian in the 1570s, both meaning "of the region around Sparta." So English had the place word first and quickly used it for the manner of speaking.
The shift to "concise" came from the reputation of the people. The Spartans were said to value saying a great deal in few words, so the name of their district became shorthand for that style of speech.
The most famous illustration is a reply to Philip II of Macedon. As the story is usually told, Philip threatened that if he entered Laconia he would raze Sparta to the ground. The Spartans answered with a single word: "If."
The earliest source we have is Plutarch, who wrote around the year 100, more than four centuries after Philip's reign. He tells it in his essay on talkativeness, and Etymonline repeats it.
A tale that neat, reported that long afterward, is best read as a legend about laconic speech and not as a documented exchange. Nothing contemporary with Philip confirms it.
What the sources do support is simpler. Laconia was a real region, the Greeks did call its people Lakones, and English used related adjectives for the place at about the time it began using laconic for terse speech.
The word has a neighbor with a similar history. Lacedaemonian, recorded in English in 1709, is another adjective for Sparta and its people. Laconic shows how a region's name can end up describing a habit that its people were famous for.