The three lessons Michael learns are what is in man's heart, what it is not given to man to know, and what it is that men live by [note that Tolstoy was not under social gender constraints to use nongendered wording].
God said: 'Go-take the mother's soul, and learn three truths: Learn What dwells in man, What is not given to man, and What men live by.
Michael learned the first when Matryona softened her heart and shared her last morsel of food with Michael, even after he reduced her one and only coat to tatters. From Matryona, he learned that love dwells in human's hearts. She felt love for Michael so could not see him suffer while she stood by. This has relevance to life today because there are so many who are suffering from want of food, shelter, security and safety all over the world. Like Matryona, while one person cannot do much, one person can do one person's worth of giving love and care where it is needed, even if it is one bowl of porridge and one piece of bread.
Michael learned the second lesson when the rich man came to have his leather turned into good boots to last a year. The rich man did not know, of course, that the Angel of Death was walking with him awaiting the right moment to beckon him from life to death. Michael learned that none can know in advance the time of their death. This applies to today's life the same way it applied in Tolstoy's time, except now there are so many more potential causes of death, from machinery to addictions. We must live each day as though it will be our last opportunity to love and do good to those in need.
Michael learned his third lesson when the woman came in with her adopted twins and he saw the end results and the truth of his mistakes. The woman adopted the twins as infants and raised them as her own after the death of her baby boy and the death of the twin's mother. From her, Michael learned that humans can live without the physical care of other humans but they cannot live without the love of God. Through Michael's voice, Tolstoy explains that this is so because God is love and that while God desires humans join together in a loving, contributing unity of effort, it is the love of God that ultimately unifies because--and here Tolstoy goes full circle back to lesson one--it is that love that dwells in human hearts and causes people to have virtues like compassion, selflessness, charity.
And Matryona sighed, and said: "The proverb is true that says, 'One may live without father or mother, but one cannot live without God.'"
The above is how the third lesson is relevant to life today. Humans are incomplete without these three things because our humanity is reduced as is our divine spark reduced without them: the world is a harder, colder place when we live without these three things.