Very often a moderately well-off person would take his chance and try to change his life, hopefully for [the] better.

moderately 是副词，由形容词 moderate（中等的、适度的）加上副词后缀 -ly 构成，意思是"适度地、中等程度地"。 the 是定冠词，在这里特指"更好的情况"这个抽象概念。 the better 这个短语表示"更好的状态或结果"，其中 the 用于强调比较级所指的特定情况。