The two shared several difficult experiences together, and it is through these experiences that they became close. In addition to that, each was physically attracted to the other.
March and Grace first met when March was eighteen years of age and peddling his books and trinkets across Virginia. During their first encounter at Clements’s farm, March was captivated by Grace’s noble demeanor and beauty, and these are among the qualities that drew him to her throughout the story. She was hospitable and even offered him a meal before he met her master. March was also touched when Grace asked him to teach Prudence how to read. There was a strong attraction between the two and they shared an intimate moment when March kissed Grace before a drastic turn of events during which Grace covered up for March and is flogged right before him. He was deeply touched by Grace’s compassion. March later encountered Grace during his service and discovers that he is still drawn to her but remained true to his wife. They later on met at Blank Hospital after March was taken ill and Grace nursed him compassionately.
Essentially, March and Grace became close because of their mutual attraction and their shared experiences.