The version of the quote in the Stuart Gilbert translation is:
"It was as if that great rush of anger had washed me clean, emptied me of all hope, and, gazing up at the dark sky spangled with its signs and stars, for the first time, the first, I laid my heart open to the benign indifference of the universe."
Following his outburst with the chaplain, Meursault felt at peace in the certainty that he had lived his life a certain way, and in the end he did not allow anyone nor any social institutions to persuade him to live another way. He took solace in the fact that he had lived genuinely, beyond the social influences that "people tried to foist" upon him. He describes the priest as a "living corpse" in the sense that the priest was always concerned with a religious/institutional outlook on life, and too focused on an afterlife rather than the mortal life; Meursault felt that the priest "couldn't even be sure of being alive."
Meursault determined that he had committed a crime but not necessarily a sin. He also determined that if there was an afterlife, all he would want out of it would be to remember the life he had; thus, still focusing on his mortal life. Meursault concludes that his mortal life and his death are the only certainties he has:
That, no doubt, was all I had; but at least that certainty was something I could get my teeth into--just as it had got its teeth into me.
In the end, Meursault felt liberated by accepting the indifference of the universe. He felt comfortable in his certainty that he had accepted the certainty of life and death without social and religious ideas trying to impose extraneous meanings to those certainties. As he was indifferent to those extraneous meanings, he felt a brotherhood with the universe which was equally indifferent to him.