The famous line 每逢佳节倍思亲 is from the Tang dynasty poem '九月九日忆山东兄弟' by Wang Wei (王维). It is one of the most quoted lines in Chinese literature, especially during holidays.
grammar
每逢 is used before a time/event to mean 'whenever.' It is more literary than 每次 (měicì). Pattern: 每逢 + event/time, + result clause.