In this sentence
Don't worry about ( what / how ) others may think of you.
I think "what" is the answer, but I do not know why I cannot use "how" in this sentence.
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"How" is associated with manner, means, circumstances.
"What" concerns the object upon which the action is carried out.
What do you explain by means of this diagram?
I explain the numerical relation between A and B. (what is being explained)
How do you explain the relation between A and B?
I explain it (what) by means of a diagram (how).
Thus, as when you think something about someone, "something" is a grammatical object for the verb "think", it follows that "what" is the proper word. In this case, the manner in which people think about someone else is not a usual concept, but you can find a telling example with a different sense of "think".
(ref.) How did you think about going to the newspaper office? What prompted you to do that?
Note that you can replace "how" with "what" and you do say a different thing.