The story
See rap (“record of arrest and prosecution”). The "ride" is being taken away in the back of a police car. Already known as a saying in 1973.
Etymology from Wiktionary, CC BY-SA 4.0.
you can beat the rap, but you can't beat the ride
Today it means: You can't escape the inconvenience and/or cost of the legal process even if you are innocent and the charges are ultimately dropped or you get acquitted. Full definition →