I assumed (perhaps wrongly) that if you don't want to deal with infrastructure you'd prefer PaaS solutions like Heroku and therefore you wouldn't want to manage not only a software package like CloudFoundry but also the servers that run it.

Well, Cloudfoundry actually offers both:

When you don't want to run the servers yourself, you can use services like appfog and nodejitsu (there may be more like those), which run a CloudFoundry stack.

When you decide "Hey, I want to run it on my server(s)", then you just setup their open source stack (there's even an automatic setup process for this) on your server and you're running on your own machine.

I assume I can't do that with your service, can I?

I don't believe nodejitsu runs on CloudFoundry. The other one I know of is Stackato.

Sorry, mixed them up: It's Joyent. And yes, the third of them is ActiveState (Stackato)