BBC Radio One, Two, Three, and Four are transmitted on FM (and others). They are free to receive, and funded by the BBC licence fee. They do not carry ads.
It's probably due to the regional/cultural diversity on HN, but I've noticed that tongue-in-cheek humor and sarcasm are inevitably followed with a handful of serious replies here.
The licence fee is payable by people in the UK who use a device to watch broadcast tv.
You don't need a licence for radios. Some people don't use a tv to watch broadcast TV, and so they don't need to pay a licence, and so to those people it is free.
Obviously, pretty much nothing is free, but at some point it becomes tedious to say "free at the point of delivery, but paid for by X".
It's not illegal, you just have to announce that it's sponsored (and Wikipedia mentions that it doesn't count as a regular play, but I can't find a citation for that at the moment): http://www.law.cornell.edu/uscode/text/47/317