The origin of this phrase is from a poem by John Donne, famous English poet. (1572 - 1631) Ernest Hemingway must have copied this phrase from Donne. It refers to the bell tolling for a funeral - in the time that Donne lived, the bells rung from churches signified various things -- in this context: death. This was particularly pertinent in his time, with the Black Death ravaging England. When Donne was writing this, he may have been saying, and it is not totally clear, that we are all going to die, and, therefore, that we need to be ready. It could actually be for us in the next instant.
However, it also seems, in the context of the rest of the passage, that he was saying that human beings are all interconnected. If one dies, so, in a sense, do we all.