This quote comes at the beginning of the final stave, and Scrooge has spent time with three different spirits who represented his past, present, and future—as well as the spirit of his old friend Marley. Each spirit has endeavored to provide lessons to the miserly Scrooge which will help transform him into a different man.
Scrooge is still wet with tears from pleading with the final ghost, the one who showed him his own tombstone. Scrooge has just finished begging this final spirit to acknowledge that it is possible for him to "change these shadows [the ghost has] shown [him], by an altered life." The ghost dissolves, and Scrooge is back in his own bedroom.
And he is relieved.
Scrooge is convinced that he can both alter his future and become the man he needs to be:
The shadows of the things that would have been, may be dispelled. They will be. I know they will!
In his relief, hope begins to grow as he starts to consider a new future. This is where his diction falls into this string of similes:
I am as light as a feather, I am as happy as an angel, I am as merry as a schoolboy. I am as giddy as a drunken man. A merry Christmas to everybody!
Scrooge is as light as a feather because he is alive and has hope that he can alter the future which the final spirit showed him. The weight of this enormously burdensome image has been lifted. He also notes that he is as happy as an angel, both alluding to the supernatural forces which have visited him and connoting a new sense of faith and goodness that the Scrooge at the beginning of the story would have never hoped for. The image of being as merry as a schoolboy asks the reader to recall the young Scrooge early in the story who was visited by his beloved sister and whose heart overflowed with love upon seeing her. It is also a reminder of the relationship with Fanny's son, with whom Scrooge intends to forge new bonds.
The quote shows Scrooge's transformation into a man who is ready to impact his world in positive ways; he has left behind his greed and negativity.