"...And this is going to be a knockdown, drag-out. And we're ready,” Harris told media outlet TheGrio in an interview released Sunday, when asked how Trump promoting birther conspiracies signals the tactics his campaign will use.

你的引文不应该只用半个引号。（你的问题，不可能是原文的问题）

when asked=when Harris was asked

how...引导的名词从句作ask的宾语

Trump promoting birther conspiracies（动名词作主语，Trump为promoting事件的主语）  signals（谓语动词）  the tactics （宾语） his campaign will use.（the tactics的定语从句，即the tactics that his campaign will use）