The firefighters managed to rescue the family that had been trapped on the upper floor of the burning building.

been 是 be 动词的过去分词形式，在这里与 had 构成过去完成时，表示"已经被..."的状态。 trapped 是动词 trap 的过去分词形式，trap 本意是"设陷阱捕捉"，这里引申为"被困住"的意思。 been trapped 表示"已经被困住"的状态。