Seen within the context of "The Death of Ivan Ilyich," it would seem that this statement would allude to the capacity, self-awareness, and willingness of people to adopt course-corrections over the manner in which they go through life. To be truly living, in this sense, is to be actively engaged in this process, while being willing to enact any changes that would be necessary for achieving true happiness and self-fulfillment. In this sense, the tragedy of Ivan Ilyich is that he learned this lesson far too late, only at the very end of his life.
Ivan is characterized as a person who values and pursues materialism and status above all else, qualities that he mistakenly believes should constitute a good life. In the process, his familial relations suffer and become strained, and (as he learns as his sickness advances) what achievements he had throughout the course of his life will not amount to anything meaningful in the face of his death.
At the same time, note how the story specifically ends, with Ivan Ilyich seeking to ensure his family's peace of mind as they face his death. He asks for their forgiveness and shows a desire to release them from the suffering his slow death has caused them. As he dies, he is described experiencing a state of light and joy, and (in this sense) his final actions do seem to bring him into a state of redemption. Thus, even if his reform only came at the very end of his life, that it came at all remains deeply significant in shaping the manner in which he finally faces death.