After climbing five flights of stairs, he was puffing heavily against the wall.

puffing 是动词 puff 的现在分词形式。puff 本意是"急促地呼吸、喘气"，常用来形容剧烈运动后呼吸急促的状态。puffing heavily 表示"大口喘气、气喘吁吁"的样子。