“Behold all lives are mine. As the life of the father, so also the life of the son is mine. The one who sins, he will die.”
The use of the word ‘soul' for nephesh in modern translations is misleading. In Ezekiel's day the philosophical conception of ‘the soul' did not exist. The nephesh was rather the life principle within him, the essence of what a man was. God had breathed on man and he became a living person (Genesis 2:7). Thus man had life because God had given him it, and that life could be taken away. As in most parts of the Old Testament, Ezekiel says nothing about an afterlife.
So here the emphasis is on this fact that man has life because he has been given it by God, that he is accountable for his own sin, and that if he does sin he will die. The wages of sin is death (Romans 6:23), but it is each for his own sin.