It is difficult to know exactly why Gurov falls in love with Anna, but we are given a number of reasons. First, he is dissatisfied with his own wife. In general, too, he is attracted to women and they to him, and he has had a number of affairs. Because he feels ashamed that he has betrayed his wife, he projects that shame on to women and calls them the lower race. This is an important detail because it suggests he has a tendency to project his own emotions on to women.
In this case, however, it is her innocence that attracts Gurov to Anna. He is about forty, but she is just out school, has not been married long, and seems a bit shy to him. She is also pretty, with a white neck and gray eyes. He find there is something touching or pathetic about her. He also assumes this will simply be a short Yalta fling and that then they will both go their separate ways.
Unfortunately, the love goes deeper for Gurov. He can't forget Anna, so he seeks her out in her hometown and they resume their affair. Like him, she is unhappy in her marriage. One day, as he embraces her while she cries, he sees himself in the mirror and is shocked at how old and gray he has gotten, while she is young. He decides that, ironically, given his age, he has fallen in love for the first time. He decides that they are kindred spirits who were fated for each other. He feels tenderness towards her and that they have forgiven each other everything in their pasts.
Gurov offers many reasons for loving Anna, and there's no doubt his own sense aging contributes to his desire to hold on to a younger woman, a symbol of youth. He also projects a sense of his pity for himself on to her. In the end, however, for all his intellectualizing, there is something inexplicable in their love, as in all romantic love, and the reader can only hope it is real and will last.