The sentence you marked in blue means that when comparing how China and Vietnam have progressed in the 30 years since the war they had with each other, it seems that China, the big mainland country, could learn some important things from the little country near it (the "little brother"), which it at one time tried to push into a lower status as a nation.
"To put into its place" is a phrase meaning to lower another person, or nation, into the lower social or power status that you believe fits its size, or education, or abilities.
The idea is that China thought because Vietnam was so small, it should be quiet and dependent as a nation, and when Vietnam got militarily powerful, China fought with it to subdue it into a quiet small nation again. Since that war 30 years ago, it seems that Vietnam has been successful as a nation in ways that China could learn from.
I hope this helps to make it more clear.