He waved his hand expansively while describing the grand vision of the entire project.

expansively 是由形容词 expansive（广阔的、豪爽的）加上副词后缀 -ly 构成的副词形式。expansive 来自动词 expand（扩展、扩大），所以 expansively 表示"豪迈地、大气地"这种扩展性的动作方式。