A scapegoat; one who takes on the responsibilities or workload of others. Here 'fall' is used with the criminal slang meaning of 'arrest' or 'period in prison'. More recently, it has also come to mean a person who is easily duped or outmanoeuvred.
It can come as little surprise to hear that this is an American phrase. It emerged around the beginning of the 20th century. The earliest citation that I can find is from the Oakland Tribune, December 1904:
"Bard is worked as a 'fall guy'. When some one comes along with a pull on Perkins and asks for a job that the senior Senator doesn't want to give him, it is very convenient to pass the burden of refusal on to Bard. ... it is easy enough to see how handy is a good 'fall guy' for cases of annoying emergency."
The term 'fall money' was also in use in the USA from the late 19th century onward. This was a stash of money put aside for a criminal while in jail. A 'fall guy' who had 'taken the rap' for a wealthier or more powerful colleague could expect to receive 'fall money' on release from prison.