It means to argue with an argument in the article you might be reflecting on. Or maybe on an idea you are arguing against.
Explanation:
An example of this would be an article about kids paying to much attention to electronics, someone might want to cut of everyone's internet so people would be more focused. A counter argument would be an argument against the reason to cut off peoples internet. Such as, kids might be highly entertained and kids enjoy doing it.
If the explanation is shaky sorry I tried my best!
