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.