It's no big surprise that death would be a central theme in a story called The Death of Ivan Ilych. The narrative centers on Ivan Ilych's struggle with a terminal illness. His struggle forces him to confront the inevitability of his own death. Before becoming sick, Ivan Ilych had willfully ignored his mortality, just like everyone around him. Once he realizes that he is actually dying, however, he no longer can. And death changes everything. Ivan becomes isolated from he world around him. All that seemed to matter before stops mattering. Ivan's death forces him to painfully re-evaluate his own life, illustrating Tolstoy's belief that only death can reveal life's meaning.
When Ivan Ilych says, "Death is finished," he is referring to the spiritual death that has characterized his whole life.
In The Death of Ivan Ilych, death is connected to life because it forces one to make an accounting of one's life before God.