What it means
The official yes that lets something start, borrowed from the traffic signal that means go. When a plan gets the green light, the waiting is over and everyone can crack on with it.
Usage examples
"The film finally got the green light after years stuck in development, cameras roll in the spring."
"The board finally gave the project the green light after months of waiting."
"Once we get the green light from legal, we can launch on Monday."
"We were all set to book the venue, just needed the manager's green light first."
"If finance gives us the green light today, we're cracking on tonight."
Where it comes from
This one comes straight from traffic lights. Green means go, so by the mid 20th century English speakers were already using green light as a neat metaphor for official approval. From there it stuck in business, politics, film, and everyday chat whenever something gets cleared to start.