Raspe was born in Seefeld in Tirol. He was described as gentle but had difficulty communicating with other people. His father had said that he couldn't stand violence. Although living in East Berlin, he was in West Berlin when the Berlin Wall was built in 1961, and stayed there, living with his uncle and aunt. He co-founded Kommune II in 1967 and joined the Red Army Faction, also known as the "Baader-Meinhof Gang", in 1970. He once said to fellow member Beate Sturm that he joined the RAF because he wanted to overcome his difficulty in relating to and making contact with people other than his family.
On 1 June 1972, Raspe along with Andreas Baader and Holger Meins had gone to check on a garage in Frankfurt where they had been storing materials used to make incendiary devices. Raspe had gone along as the driver (they were driving a Porsche Targa). However as soon as they arrived at the garage, police began to swarm around the scene. Meins and Baader had already entered the garage and were surrounded but Raspe, who had remained by the car, fired a shot from his gun and tried to run away when he was rushed by police, but to no avail; he was caught and arrested in a nearby garden. Meins and Baader were arrested soon after.
Raspe was convicted on 28 April 1977 and sentenced to life imprisonment. On 18 October 1977, Raspe was found with a gunshot wound in his cell in Stammheim Prison, Stuttgart. He died shortly after being admitted to a hospital. Fellow RAF members and inmates, Baader and Gudrun Ensslin, were found dead in their cells the same morning. Irmgard Möller was found in her cell, wounded after supposedly stabbing herself in the chest, but survived. The official inquiry concluded that this was a collective suicide, but again conspiracy theories abounded. However, none of these theories were ever brought forward by the RAF itself. Some have questioned how Baader managed to obtain a gun in the high-security prison wing specially constructed for the first generation RAF members. Also, only a total commitment to her cause could have allowed Möller to have herself inflicted the four stab wounds found near her heart. There are many debatable aspects to the deaths: Baader was supposed to have shot himself in the base of the neck so that the bullet exited through his forehead; repeated tests indicated that it was virtually impossible for a person to hold and fire a gun in such a way. In addition, three bullet holes were found in his cell: one lodged in the wall, one in the mattress, and the fatal bullet itself lodged in the floor, suggesting that Baader had fired twice before killing himself. Finally, Baader had powder burns on his right hand, but he was left-handed. Raspe, however, showed no signs of powder burns, sympathizers and Irmgard Möller persist that the deaths had been extrajudicial executions.