Following his successful TV appearance, the author was flooded with offers from publishers.

was 是 be 动词的过去式，表示被动语态。 flooded 是动词 flood 的过去分词，原意是"洪水泛滥"，这里用作比喻，表示"大量涌入"。 with 是介词，表示"伴随"或"带有"。 was flooded with 这个短语字面意思是"被...淹没"，比喻"收到大量..."，表示收到数量多得惊人的邀约。