As @hansen stated, you're not buying anything other than an automated installation service from the site you're referencing. They essentially deploy a Droplet with DigitalOcean, run an automated installer and then hand things over to you -- that's the general gist of how it works.
That said, from their wording, I can see how one would be confused. Also, despite what they state on their site (i.e. anonymous IP) -- the IP isn't anonymous. It can be traced back to DigitalOcean, as could any IP to any provider as ARIN/RIPE make this information public.
That said, me personally, I wouldn't exactly trust any site that claims that a VPN will provide anonymity or censorship avoidance as it's simply not true -- whether it's $3 or $300.
A VPN will encrypt traffic, yes, though it isn't meant to provide 100% anonymity or prevent censorship. Where there's a will, there's a way. If someone higher up, somewhere, wants to monitor you, they can and will by any means necessary. Does that mean they will? No, not at all. I'm just stating that a VPN isn't going to protect you 100% from everything.