Dimitri Gurov's "two lives" relating to Anna don't start until after they have both left Yalta, so you are asking about the portion of the story where Chekhov describes Gurov's psychological character development. In the first part of the story, Gurov also has "two lives" but relating to his repeated infidelity to his wife whom he disliked and of whom he was afraid:
[His wife] was a tall, erect woman with dark eyebrows, staid and dignified, and, ... and he secretly considered her unintelligent, narrow, inelegant, was afraid of her, .... He had begun being unfaithful to her long ago -- [and] had been unfaithful to her often ...
At first, he had every expectation of this "adventure" with Anna being just one more in a series, yet he felt remorse at her departure from Yalta. Chekhov connects this remorse with Gurov's sense of having "unintentionally deceived her" in his representation of himself: she thought him kind and good, he was being coarse, ironic, and condescending with her: "there had been a shade of light irony, the coarse condescension of a happy man who was ... twice her age."
It may be the nagging knowledge that though she loved him she had not been happy with him, like the previous women had been ("who loved cheerfully and were grateful to him for the happiness he gave them"), that provided the contradictory psychological impetus for the changes that came over Gurov. The first change that he saw within himself was that Anna's memory did not fade but seemed to grow bolder and brighter. She was suddenly in his thoughts at any moment; he dwelt upon her memory during mundane moments of living; her image haunted his dreams and became intertwined with his future: "[she] followed him about everywhere like a shadow and haunted him ...." The only way to fulfill his "intense desire" was to see her again.
Gurov was willing to undertake this new kind of "two lives" because he had an epiphany that Anna truly loved him as he felt her shoulders shaking with emotion. This realization of her true love inspired in him the desire to be "sincere and tender," to love her in return; to recognize that they loved each other "like people very close and akin, like ... fate itself had meant them for one another."
Gurov seems to be motivated by negatives, by things that remain as puzzles or contradictions in his mind. For instance, Anna was not happy with him, yet she loved him, presenting a puzzling negative contradiction to his mind that led him back to her. Earlier, he had been insincere yet she thought him good and kind, presenting the first negative contradiction. Finally, her weeping led to his realization that he wanted to be genuine and love her in return. This is how Gurov might be explained and these are the things he saw in himself that made him understand he loved Anna. Yet, since these are all built upon negative contradictions, the question remains of the enduring strength of Gurov's psychological changes of heart.