Even though it is never specifically stated as to why the little boy didn't go home after leaving the temple, a close look at the text of this story itself can reveal his reason for choosing not to return home.
The boy is originally from a farm, and when he was growing up, he didn't fit in there. At the beginning of the story, he is described as "quite weak and small, and people said he could never grow very big." Because of this he "did not seem to be fit for hard work." On the farm, he would be a much better help if he were big and strong, but instead he is clever and spends his free time drawing. He initially left because he was nothing but a burden on his family.
After the experience in which his cat drawings kill a large, deadly "goblin rat," he vows to never stop drawing cats. He felt compelled to draw them before the experience, which is why farm life didn't suit him. The experience with the goblin rat only validated his artistry, so it makes sense that he would continue through the world until he found a place where he could fit in and draw cats in peace. At the end of the story, the reader is told that he "became a very famous artist," so it is clear that he eventually found his place in the world, and it wasn't the place where he was born.