To answer this question, we must think back on the plotline about Elya Yelnats and Madame Zeroni. Elya was in love with Myra Menke and wanted to earn her love in return. When Myra's father says he would rather have a fat pig than a boy with a heart full of love for his daughter, Elya was very sad. His friend, Madame Zeroni (Zero's great-great grandmother), gives Elya a small pig and tells him to carry it up the mountain each day, to let the pig drink from the stream, and to sing a magical song each day.

Elya followed the instructions and carried the pig. As the pig grew, Elya became stronger. In the end of this sub-plot, Elya forgot his promise to carry Madame Zeroni up the mountain, sing the song, and let her drink from the stream.

Stanley carrying Zero up the mountain is a parallel plot. Like his great-great-pig-stealing grandfather, Elya Yelnats, Stanly has gotten strong by digging holes every day since his arrival at Camp Green Lake, similar to how Elya became strong by carrying the pig. Stanley is a large boy and Zero is smaller to begin with, but Zero is also sick and weak. He is dehydrated and has lost weight.

It is easy for Stanley to carry Zero because he is strong from digging holes, Zero is small and light, and Stanley is fulfilling his great-great-grandfather's promise by carrying Madame Zeroni's descendant.