The problem is widely political by my experience. I had a similar issue when SO was not blocked, but forbidden by customer policy. Yes, customer. In fact we all were working by connecting our laptops to customer's VPN and opening a remote desktop to their development machines. Disconnecting and reconnecting often was not only a waste of time, but required justification.
Those didn't have (yet) a block on SO but once I got called from my company's upper management reporting that customer code was found, despite well redacted, on SO. Question has been deleted years ago for this reason despite a good answer.
All politics. When you work as a consultant the customer pays your company in order for you to solve their problems, not for Googling around. And despite Google-coding is deemed a best practice in many environments (it best matches secure coding), some customers are still old-minded and paranoid.
In my case there might also have been a privacy law violation (in my country firewall records can be accessed only by police officers under a court order for criminal investigations, not upon sysadmin request or regular scan), but more simply the customer could have crawled (as it was told me) SO to find their own code in the questions/answers.
Comments apart, let's go to the answer. This answer is meant for all the people who face the same problem as the OP.
Before trying to convince your manager you have to understand if this policy was enforced because of a practical, tangible and legitimate reason. You should first set a meeting (or write your manager an email) requesting why SO is blocked and to unlock it because you need to use it for legitimate work reasons.
That mail, depending on the size of your company and the cooperativity of your manager, could be escalated to sysops. In the best case you will get it unlocked in a few hours, because it might have been set by firewall's default policy and just nobody bothered before. This applies to companies whose core business is not IT but have internal IT staff (e.g. you are the sole developer the fashion house's storefront). But if you work in a real IT company I couldn't believe nobody noticed SO blocking before....
If you don't get a good answer you should confront your manager, politely, to discuss the issue possibly verbally, in a meeting or even in front of coffee machine. Be firm and serious about your will to discuss the issue and get a complete answer. Highlight, for example, the great productivity benefits by pointing links to questions that have been answered already. This will prove that SO saved you a certain amount of worktime (read it company money) and increased your productivity by hundreds percentile. Speak by numbers. @JorysMeys and @Nicolàs pointed this out in their answer.
In the worst-worst case you may get an answer that "we pay you, not them, to solve problems". In that case you will discover the politics behind this policy and probably want to send your resume to my company :-) which promotes Google-coding and SO-coding having an SO moderator* in their staff :-)
On the contrary, you may also get a security-related answer I heard by experience. "Who audits the code posted by anonymous community members?". The answer is: yourself. Sometimes people don't understand that SO is made to solve narrow problems. Every programmer must not copy&paste code from SO, they must read and understand it. Quality answers provide explanation, not just code. And the best answers provide less than 5 lines of code. And since we in SO redact your code both to hide references to existing people/companies, both to make it applicable to a generic case, that is another good reason to prove that no special code audit should be done to code provided by SO community members. It is developer's duty to assemble the knowledge acquired from SO or other forums into a working solution.
Happy new year!
*I am not community-elected moderator, I just have enough rep to "access moderator tools"