How can we still see the cosmic background radiation when its source has vanished, and all radiation has to have a source?

For the same reaon that I can bradcast a radio signal into space, and then dismantle the transmitter, with the signal still moving outwards at the speed of light. Electromagnetic radiation, once produced, is what we call 'self- regenerating', and will continue to travel outwards from its source at the speed of light until it is absorbed. With the Big Bang, this radiation still fills all of space and will continue to do so forever if the universe is infinite.