What it means
You call something open and shut when it's so obvious the argument barely gets out of bed before it's over. Most often it's said about police stuff, court cases, or any mess where the proof is sitting there loud as day. It can also be used for everyday situations that feel basically settled already.
Usage examples
"The referee call was open and shut, even the losing fans agreed."
"With his fingerprints on the safe, it was an open-and-shut case."
"With his fingerprints on the till, the police called it an open-and-shut case."
"It looked open-and-shut at first, but the new witness turned everything upside down."
"Come off it, mate, that's open and shut, he literally texted the confession."
Where it comes from
This comes from legal English. An open-and-shut case was a case so clear it could be dealt with fast, with little room for argument. The phrase is recorded from the late 19th century and later spilled out of courtroom talk into everyday English for anything that seems dead obvious.