After Starbug was destroyed, the future crew no longer existed - therefore were unable to go back in time and kill the present crew, hence they survived.
But unfortunately disaster has struck. Starbug is completely devoid of curries. Lister proposes that they go back in time to order a couple of hundred curries from an Indian Restaurant. The rest of the crew will not go back in time because they are afraid of becoming the future selves they saw. However, Lister removes a guilt chip from one of Kryten's spare heads and swaps them over, telling the new guilt-less Kryten to reassure them it will be OK.
They go back and appear in the Texas School Book Depository in Dallas on November 22nd 1963 just as Lee Harvey Oswald is taking his shot at President Kennedy. They knock him out the window where he dies hitting the ground, preventing Kennedy's assassination.
To avoid being captured they go ahead a couple of years, however Kennedy's survival causes an alternate reality where he has been impeached out of office and the new president is controlled by the Mafia, allowing the Soviet Union to build several Nuclear Silos in Cuba. Fearing the Soviets will start a nuclear war, all the major US cities are deserted.
Lister goes to the airport which would have been renamed JFK Airport where Kennedy is being transported to prison. He persuades Kennedy to go back in time with them, and be the gunman on the grassy knoll, and shoot himself for the ultimate conspiracy theory. After restoring Earth's normal timeline, The gang return to Starbug. Of course they still haven't got any curries...