If you own a bar and you own a CD, you are allowed to play your CD in your bar. The article is pure idiocy. Bars don't need to pay licensing fees.
17USC106:
Subject to sections 107 through 122, the owner of copyright under this title has the exclusive rights to do and to authorize any of the following
[...]
(4) in the case of [...] musical [...] works, to perform the copyrighted work publicly;
(emphasis mine).
17USC101 defines public performance:
To perform or display a work “publicly” means— (1) to perform or display it at a place open to the public or at any place where a substantial number of persons outside of a normal circle of a family and its social acquaintances is gathered
it also defines perform as:
To “perform” a work means to [...] play, [...] it, either directly or by means of any device
So, playing your CD in a place that is "open to the public" is performing it publicly, and the copyright owner has the exclusive right to do that. This means that if you want to do it you need a license from the copyright owner.
Note also that a recording of music typically has three distinct copyrights on it. (1) The songwriter's copyright on the tune, arrangement, etc., basically everything you'd find in the sheet music other than the lyrics, (2) the songwriter's copyright on the lyrics and (3) the recording artist's copyright on the recorded performance. It's not uncommon for there to be a lot more than two songwriter copyrights, and in the case of recordings that contain significant sampling, there can be more copyrights in the recording, too.
To play the CD in your bar, you need licenses from all of the copyright holders. As others have mentioned, the record labels take this seriously and there's a high probability that infringing their copyrights this way will result in your being sued for millions of dollars, because the law authorizes statutory damages of up to $150,000 per offense.