Question: I was recently speaking with my manager, who stated that while my work is satisfactory, they've been noticing I'm spending a lot of time of Stack Exchange. I find Stack Exchange to be very informative and helpful in my work.

Particularly considering this Stack Exchange site has many users who spend a large portion of their time on here, how can I explain to my manager how useful Stack Exchange is for me?

Answer: 
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> how can I prove its value?
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You can't because it doesn't have any in that context. If you're using it to solve problems once in a while that's one thing, but if you're 'spending too much time' on it, then that is clear heads up from your boss to cease and desist.

Your boss isn't attempting to open a dialogue about pros and cons, he/she is warning you to spend less time on StackExchange.
You can't assume that the benefits are self-explanatory. You need to do as  

@JoeStrazzere suggested in the comments and detail the specific things you have learned that are useful for your specific job. In particular, detail what work-related problems that you found solutions for on Stack Overflow. If you can show that you spent less time solving problems by looking for them on Stack Overflow, then that would be a good thing as well.

You also need to prove that you are not be behind in your work. No manager is going to support someone spending a lot of time on the Internet at any site, if the work is behind without specific proof that the time was spent researching the specific issue(s) that was(were) making your work delayed. Further, if your coworkers are solving their problems without spending hours on Stack Overflow, then it is best to bite the bullet and spend time here only when researching a specific issue.

If you are learning new things that are not immediately useful, then I would suggest you do that in your off hours.