I thought she would listen, but I should have known better.

known 是动词 know 的过去分词形式，know 的意思是"知道、了解"。 better 是形容词 good 的比较级形式，意思是"更好的、更明智的"。 known better 是一个固定搭配，字面意思是"知道得更好"，引申为"应该更明智、应该更清楚"。