These people often carried with them memories of his politen

carry是及物动词,直接加宾语,不需要加with
这里的with them是在他们身边的意思,
如果按你理解,他们的记忆,
则是carry their memories,不要加with
句意：这些人们总是把他的礼貌他的好的记忆带在身边
句子也可以把with them放到后面,变成
These people often carried memories of his politeness and fairness with them .
这样你就理解了吧
再举一句例子
我总随身带个包
I always carry a bag with me.
= I always carry with me a bag.