These announcement typically only intended for downstream providers (regional Iraqi ISPs in this case) but sometimes "leak" upstream - erroneously announced to the open internet which sometimes cause outages for the whole world. This is what famously happened with YouTube when Pakistan tried to do the same thing in 2008.
In this case the routes "leaked" upstream again like what happened before but seems like outage was mostly prevented by something called RPKI which is basically a technology to attest who really owns which prefix.
And yes. It is a problem. If your upstream does skulduggerous things, you can't "route around it" from the standpoint of being an endpoint. Your packets will go where your ISP says they go.
Unless...
You take a bit of the routing decision out of their hands, which takes a bit of footwork on your part. For instance, setting up a VPN to a network zone unpolluted by the faulty prefix announcement, which is basically going to be any non downstream of the hostile ISP provider.
Once you're out of that routing zone, normal network visibility is restored. Odds are even a national scale backbone provider is not going to be able to effectively block traffic that's routing out to a proxy, so all the the ISP has really done is made life more difficult for people unaware of how to set up such an arrangement.
Which now that you know about this, it is your duty to spread the knowledge of how to do so far and wide. If someone wants to block it, then that's all the justification needed for frustrating those efforts.
Today's highly-centralized server-client Web architecture does not have this property. Some P2P protocols are closer to the spirit of this quote.
Conclusion: architecture of "the Net" matters a lot (the original quote didn't use the term "Internet").
The quote has nothing to do with automatically and invisibly routing around censorship, e.g. a techno-system that somehow always works to oppose censorship. It's just that people will reconnect stuff eventually, bypassing any block somehow. At worst there are always sneakernets.