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.
I think your problem is not so much about proving the value of stack exchange, but about justifying your productivity to your manager.

There are two scenarios to consider here:

1. You are **under-performing** (i.e. you are not meeting deadlines or whatever output you are producing is of poor quality) then maybe you **do** need to look at how much time you are spending on these sites, at the very least to give your manager the impression that you are making an effort. If you are regularly checking these sites for updates to questions/answers, you may find that these regular interrupts are actually reducing your overall productivity.
2. You are meeting your deadlines and otherwise producing quality output. In this case maybe you should discuss this further with your manager. You don't want them to have the perception that you are slacking off, and it may help to explain to them that the information you glean from these sites actually increases your productivity as you are not always re-inventing the wheel.

In either case, it is important for you that your manager does not perceive that you are goofing off on work time. If nothing is forthcoming from these discussions, it may be politic to limit your usage to break periods.