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.
So there are two possibilities here - either your boss is over managing and just want to show you that it's he who is in charge here and you should obey or - or - he is not satisfied with the quality of your work. 

There's no other option in first case than to leave - never waste your time working for non-professionals. But to be honest with you the very fact you had asked this question and the way you asked the question hints that you assume that your boss is a reasonable person who can be convinced in something if he was wrong. That said, we don't investigate the first possibility here. 

In second scenario there's only one way to convince your superiors that you are doing it right - concentrate on your work. Words like "we are more or less satisfied, it's just that you spent too much time surfing Stackexchange" could be a sign that they are actually not satisfied and politely indicating this. Because actually good boss not that much care what and when you are surfing in case the work is done. 

Here's the strategy I suggest to stick to:

* Sort tasks you are going to work on today by how easy you can solve them without Internet, relying on your own knowledge, do the easiest ones first.
* Don't surf Stacks, don't read interesting questions and answers even if they are related to you work in that sense that they are making you a better professional overall. Don't answer, don't comment. Just google an answer to your particular question and, as soon as you've found it, paste it to a text file or (even better) your corporate wiki.
* Make a habit to first look for solutions in that local storage I've mentioned earlier. You'll be surprised how many routine repeating stuff can be saved and easily find in a simple text file without any googling at all.

I encourage you to start with these steps and trust me, you'll see how your work relationships will get better.