Why does Chapter III of The Stranger end with "the blood pounding" in Meursault's ears and "the dog whimper[ing] softly"?
Different translations may use different synonyms for "pounding" and "whimper softly."
The answer to this question is a complex one partly because of Camus' innovative psychological style and partly because of symbolism and foreshadowing. Firstly, Camus believed that the psychological aspects of a psychological novel were revealed in action, which ran contrary to the popular psychological literary convention of his day that built psychological development through first-person introspection and monologues of self-examination. What this means is that in order to show Meursault's psychology and psychological development, Camus develops actions, which includes dialogue and first-person descriptions of passing moments--but not passing thoughts.
Secondly, at least one recurring symbolic motif is present in the passage you ask about:
The whole building was as quiet as the grave, a dank, dark smell rising from the well hole of the stairs. I could hear nothing but the blood throbbing in my ears, and for a while I stood still, listening to it. Then the dog began to moan in old Salamano’s room, and through the sleep-bound house the little plaintive sound rose slowly, like a flower growing out of the silence and the darkness.
The symbolic motif I'm thinking of is flower. There are at least four places in which Camus likens something to a flower, including Marie's face: "her sun-tanned face was like a velvety brown flower." The first use of flower is when Meursault sits in vigil at his mother's coffin:
through the open door came scents of flowers and breaths of cool night air. I think I dozed off for a while.
I was wakened by an odd rustling in my ears.
The flower symbolizes the existential absurdity in life, the flower appearing as it does in a completely random and valueless fashion (1) at the side of death, (2) in a poorly lit and ill-smelling stairwell and (3) in a beautiful woman's face. When, as in the first two uses (vigil, stairwell), flower appears with the symbolic motif of sound rushing in Meursault's ears ("odd rustling in my ears"; blood throbbing in my ears"), existential absurdity is linked directly to Meursault's life, which leads to the next consideration, that of foreshadowing.
Thirdly, the scene set by Camus in the ill-lit stairwell--
"quiet as the grave, a dank, dark smell ... blood throbbing in my ears ... dog began to moan ... through the sleep-bound house the little plaintive sound rose, like a flower growing out of the silence and the darkness"
reflects back on the earlier scene of the vigil at the coffin echoing the funereal motif of death and decay, with the dog moaning as mourners may do and as Meursault perhaps ought to have done at the vigil. This foreshadows what will eventually happen to Meursault largely as a result of the overwhelming influence of his behavior (or lack of behavior) at the funeral vigil.
So why does Chapter III of The Stranger end with the "blood throbbing" and the "dog moaning"? It ends thusly to express Camus' existential views; to tie random meaninglessness with Meursault's life; and to prepare the reader for--to foreshadow--the events that will ultimately follow; and to prepare for the great and significant influence the funeral vigil has on Meursault's ultimate end.
Mersault is feeling alive for the first time in his life. He is overwhelmed by the unexplained bloodlust he has just experienced. The dog is either whimpering because he is afraid of the crazed attitude of Mersault in this state or as a symbol of Mersault's comparable power at that moment. Or both.