What does it mean when a television station is state-sponsored?

It is probably controlled by the country it is produced in.

Not always is the subject matter approved by the State, although since the station is State-sponsored, there is always the expectation that the producers, directors and writers will self-censor.

PBS is State sponsored. It means the State provides some or all of the funding for the television station or network.

The state could endorse what appears on the station, but it is not always necessary.

I agree with @WestRiverrat.

It means the station does not have to pander to advertisers, meaning the programmes can be more honest. Programmes like Top Gear UK, where they actually can honestly rip into the cars and criticise them properly would be impossible on private television. You can rip the godawful corvette only once before GM drops its sponsorship and advertising on the station.
Because the money a private station gets depends on the number of viewers, a public station does not have to rely on high ratings, and thus can have more quality programmes, instead of programmes that bring high ratings. More educational programmes instead of Beavis and Butthead.

BBC is a good example. It is State funded, but it has a charter to be run under a certain set of rules that ensure its independence. It has been questionable and tested from time to time in its history. http://www.bbc.co.uk/bbctrust/about/how_we_govern/charter_and_agreement/index.shtml

Here is the Norwegian Broadcasting company bylaws in English. They are publicly funded and their existence is guaranteed regardless of the waxes and wanes to the left or right in government office. http://www.nrk.no/informasjon/about_the_nrk/1.4029867

A lot of radio stations are owned by various states.

Vol State University has its own FM station and it’s owned by The State of Tennessee.

It’s all public service announcements, but they play great oldies.