如果将这句句子稍稍调整一下,你可能就明白了：
They were an elderly couple who lived on what was left of the family.(他们是一对老夫妇,住在一套破旧的房子里,靠着家属农场仅存的家产生存)
这里,on应该是live on,靠什么生存,只是中间插入了in an old house.
这样是不是好理解了一点.