My mother-in-law has been staying with us this past week and she's driving me crazy.

driving 是动词 drive 的现在分词形式，drive 本意是"驾驶"，在这里引申为"驱使、迫使"。 crazy 是形容词，意思是"疯狂的、发疯的"。 driving me crazy 是一个固定表达，意思是"让我抓狂/发疯"，形容某人或某事让人非常烦躁或难以忍受。