In the same way that I can't buy a cd and broadcast it at a radio station without paying royalties, does a musician have to buy or pay extra to perform a piece?
In the US, compositions, performances, and recordings are all handled separately.
As a musician, you are free to make a recording of any music you want, and once you do so you own the copyright on that recording. If someone else records the same music, they also own a copyright on their own recording (note this does not mean something like recording a song off the radio or copying a CD or mp3... that would be a copy of someone else's recording). What gives value to one specific recording over another is some level of "officialness" or fame... if you make the song famous or wrote the original song, your recording is the one people will want.
For a performance, a copyright license is paid through the venue... but only to the author/composer (and the music and lyrics may be registered separately). This money is paid to a collection society like ASCAP or BMI. In the case where there is a specific recording of a work that everyone knows, if you cover that song, the band that originally recorded it and made it famous it gets nothing if they used a song first written by someone else. In the case of radio... there are no license fees. In fact, not too far in the past the money went the other direction, with labels paying radio stations to play songs for promotional reasons. This is called, "payola", and more complicated schemes still take place today.
Now for compositions. In the US, a printed musical composition is not different from any other printed work. That is, the author has a copyright... only they can make more copies under the law. But just like you can read a book out loud, you can play the music in the composition anywhere and in anyway that you want. Once the copy has been made, the author's say in how you use that copy is done. You can even re-sell it if you want. There is a difference, however, when it comes to performances. As a composer, you are legally entitled to a royalty when your music is performed. In exchange for this entitlement, you lose the right to license the work yourself, which means losing the right to say, "No" if your music is used at an event you don't like. Instead, there is a compulsory license, and the money is funneled through ASCAP or BMI. If the American Nazi convention latches onto your song for some chance lyric phrase, and get their own members to create a cover recording, you can't stop them from using it as long as they stick to ASCAP-licensed venues.
It's going to depend on the laws of the country. In U.K. there are two organisations which control payment for this sort of thing - P.P.L and P.R.S. The money for performances are paid by the venue. So, in a way, the band is paying, as it'd get paid more, maybe, if the venue didn't have to find this fee. Occasionally a band which I've played with has been asked for a list of performed numbers, but as far as the musos are concerned, no payment came from them. Music that has been published and purchased already has a proportion of its sale price assigned to the writers, so no extra charges there, either.
I can only speak for opera but in that case there is a thing called 'Grand Rights', which are payable if you perform an opera any of whose authors (i.e. including librettists) are still in copyright, i.e. in our case died less than 70 years ago. These are rather large and at the publisher's discretion, and typically involve a piece of the gate.
Contrary to the statements elsehwere here about the venue being liable, in our case the opera company was liable.