How does Scrooge's recollection of his childhood bring about his reform?
For a moment in time, Scrooge is able to see his past and see what he used to be. It provides a moment in time where he is able to see the transformation of his being from where to where has come. For this reason, it is powerful in its transformative effects. For example, Scrooge sees himself as a child, apart from others, but armed with a vibrant imagination, one that regales of adventure from books like "Ali Baba" and Robinson Crusoe. In this recollection, Scrooge is able to see what he used to be, a vision that is so vastly different from what he is now. The power of the intangible imagination and sense of adventure that lit his soul as a child is replaced with the "cheap darkness" that envelops him now. The recollection of his childhood makes this evident to Scrooge. Scrooge's recollection enables him to see a point where he was different and where he used to be so much more on an intangible level from what he is now.
At the same time, Scrooge's recollection of his past makes him realize the people in his life that used to embody this intangible goodness. This recollection helps to bring about his reform because he has done a good enough job of cutting this part of this past from his being in the world. In recollecting the joy and selflessness that Fezziwig and his wife embody during the holidays and in their dancing, Scrooge realizes how far he is from such a vision of being in the world:
He has the power to render us happy or unhappy; to make our service light or burdensome, a pleasure or a toil. Say that his power lies in words and looks; in things so slight and insignificant that it is impossible to add and count 'em up, what then? The happiness he gives, is quite as great as if it cost a fortune.
When Scrooge is able to see that "his power lies in words and looks," it is a statement that reminds him about how far he has strayed from such a reality himself. As he couples this realization with Belle's rejection, it becomes evident that this recollection of his childhood and past help to fuel Scrooge's reform. They are not the deciding elements, but in seeing his past, especially his childhood in light of the present, Scrooge realizes how lost he truly is. This becomes confirmed with other elements of his present and future that cement his reformation and eventual redemption.