This phrase was used when a person addressing other people said:
You went to school on us
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Divide the sentence into two parts:
Our enemies went to school ...
means our enemies learned.
... on us.
means at our expense, i.e. we provided the means for them to learn.
"They studied our doctrine, our tactics, our equipment, our organization, our training, our leadership," he said. "And, in turn, they revised their own doctrines, and they are rapidly modernizing their military today to avoid our strengths in hopes of defeating us at some point in the future."
He's saying they were able to learn by studying the example we have provided.
In that context does it mean that they copied us?
No, it means they have learned how to operate against us, using their own methods.