It's often used to suggest:
* Moving from planning to action: "We've been talking about this project for months, let's get the show on the road!"
* Urging someone to hurry up: "Come on, let's get the show on the road! We're already late."
* Starting a journey or trip: "The bus is leaving soon, let's get the show on the road."
The idiom originates from the idea of a theatrical performance beginning. Getting "the show" on the "road" refers to moving the set and actors from their storage location to the stage, marking the start of the performance.