Basically, he says that when something interrupts us like how for instance if death occurs our brain and all the elements that make up the brain are gone very much like the path of a riverbed it stops flowing when it is interrupted, it ends.
The mind and soul of a person ceases when one dies. In reference to memory, memory is brain dependent. It is acknowledged that if there is injury to the brain it often results in a temporarily sometimes permanent loss in memory depending on how severe the injury is. For the reason that memory is brain dependent it is a hind to us that our memory does not continue after we cease to exist. This led Russell to believe that the best explanation of life after death is that death is the end and the there is not possibility for life after it.