A greaseman is a guy that can get you in. Also a guy that can make things happen smoothly.
In the movie "Ocean's Eleven", Yen was a 95 pound "little chinese guy", who worked as an acrobat in a Las Vegas show. He was so flexible that he could fold in half and fit in a very small tube in a portable safe. At the time of the robbery this is how he got into the vault of the casino. He got in, and therefore they called him the greaseman.