I've been using Statcounter for the past 3 years on my forum, and now on my J! site, and I've never seen anything like this, every page this visitor looked at (about 50 pages) was preceded by the bolded text above.
Looks to me as though the user may have spidered the site, downloaded it locally, and that your stats counter was still working failthfully while they were checking it out and still connected to the internet.
A program would be required - and there are literally hundreds available. It's hard to say without evidence (after the fact unfortunately) whether or not the person who did it has good intentions or bad. These things are distributed with the premise of people being able to browse websites offline - but the prevalence of broadband around the planet has kind of made that redundant.
Yes, I'd be thinking content theft too. Google unique keyphrases from your content over the next few weeks / months and see if anything comes up.
Unfortunately (because the spiders use standard HTTP requests), the ONLY way to combat it would be to put some sort of request limiter on your site - which I would consider quite difficult to do without canning legitimate requests also.
It's really just one of those things that happens out there mate. Short of sending a "cease and desist" letter if you find the culprit, you may need to be satisfied with the knowledge that "Imitation is the highest form of flattery". The trick is to stay above the bastard in your search engine rankings.