Despite our concerns about his health, my grandfather insists on mowing the lawn himself every weekend.

insists 是动词 insist 的第三人称单数形式，意思是"坚持、坚决要求"。 on 在这里是介词，与 insist 搭配使用，表示"坚持做某事"。 insists on 是一个固定搭配，意思是"坚持要（做某事）"，后面接动词的-ing形式或名词。