This short story by Chekhov concerns a student who reflects on his own life and history and initially sees nothing but despair. As he sits by a fire and recounts the story of St. Peter from the Bible, and how he denied knowing Jesus three times, it seems as if he is a figure who is overtaken by despair and sorrow. However, after recounting the story, he sees that the truth within it provokes an emotional reaction in his audience, the old woman, Vasilisa, and her daughter, and this helps him to recognise that there is a much more optimistic way of looking at life and history, which is summed up in the quote identified in this question:
...he thought that truth and beauty which had guided human life there in the garden and in the yard of the high priest had continued without interruption to this day, and had evidently always been the chief thing in human life and in all earthly life, indeed...
He recognises that the emotional reaction in response to this story is a sign of the "truth and beauty" that is present in human life throughout the ages, and this acknowledgement of the continuing existence of that truth and beauty gives him hope for his present and for his future, turning his despair in to optimism.
These lines could imply that human beings can transcend the universally-shared experience of suffering. As the student discovers that the "past...is linked with the present by an unbroken chain of events," he also comes to understand that there is always hope in life.
In the beginning of the story, the student miserably ponders his bleak existence; he feels that life will never change:
...there had been just the same desperate poverty and hunger, the same thatched roofs with holes in them, ignorance, misery, the same desolation around, the same darkness, the same feeling of oppression -- all these had existed, did exist, and would exist, and the lapse of a thousand years would make life no better.
As he interacts with Vasilisa and Lukerya, however, the student comes to realize that he's been looking at life the wrong way.
In telling the story of Peter's betrayal of Christ, the student highlights the frailty and imperfection of the human spirit. However, in acknowledging that the "truth and beauty which had guided human life there in the garden and in the yard of the high priest had continued without interruption to this day," the student also reinforces the idea of Christ's victory over despair.
In his story, the student states that Christ "went through the agony of death in the garden." The agony in the garden happened earlier in the same night that Christ was disavowed by one of his most faithful followers, the eve of his crucifixion. However, for many Christians, the crucifixion of Christ precedes his resurrection. Thus, victory and life are borne out of seemingly hopeless suffering.
In telling his story, the student is able to help the women and himself transcend the difficulties of their present existence. This is why, as he walks home, the student finds himself reveling in "the feeling of youth, health, vigor." He is seized by an "inexpressible sweet expectation of happiness," and he begins to see how life can be "enchanting, marvelous, and full of lofty meaning." He begins to understand that, just as he is linked to the women by their shared suffering, he is also linked to Christ by his subsequent victory over death and despair.