Many of the team members went down with food poisoning after the dinner.

went 是动词 go 的过去式，表示"去、发生"。 down 在这里是副词，表示"向下、倒下"的状态。 with 是介词，表示"伴随、由于"的原因。 went down with 是一个固定短语，意思是"因...而病倒"，特指因疾病或不适而倒下。