Elmo would often bragg to his cellmates about all the people he killed as well as the women he slept with. At one point, he got a job got a job bussing tables at a country club. He would look at the regulars who came to the country club to see which one he could rob. He decided on golf pro Glenn Quinton, following him to his house one night. He entered the house and Glenn yelled at him. Elmo flew into a rage and drew his .38 special revolver, the muzzle of which he wrapped towels around to muffle the gunshots he used to murder Quinton and Andy's wife. Because Andy also owned a .38 caliber revolver, he was framed for the crime.
Once he was arrested for robbery, he had bragged about this particular crime to his cell mate, Tommy Williams, who later told Andy about his crime once he was taken to Shawshank State Prison. Andy had attempted to convince Warden Norton of his innocence, but due to the Warden wanting to keep Andy from telling others about him laundering money, he refused to accept his "story".
Due to the fact that Andy does not bother to clear his name after his escape from Shawshank Prison, and Norton kept Tommy quiet about him innocent, Blatch presumably got away with the crime.