I can't agree fully with any of the answers...closest is Tony's comment

Yes, Mc Gregor is big/popular now and a working-class here so people who would not have cast him a second thought or glance in the past (when he was unknown and because they look down on MMA as a sport) now want to be associated with him and get some of that popularity to rub off on them (so politicians, for example, will hope their own popularity will improve if they are seen with McGregor, and commercial enterprises will hope they'll sell more stuff, get more people into their restaurants/clubs etc.
They are hoping to use him and his popularity to advertise themselves and their own enterprises so they will also be popular. So they "want a piece of him" means they want to vicariously get part of his advertising potential, ride on his coattails and by osmosis share in his success. In other words, they all want to get on the bandwagon! (But there is still a certain disdain for the sport and it's even questioned as to whether it's a "sport" or not)
http://dictionary.cambridge.org/dictionary/english/jump-clim...

And BTW questions were asked here as to why the Taoiseach or PM had NOT officially congratulated McGregor as usually happens when someone achieves something on the international stage as it was felt he was missing out on an opportunity to increase his own and his party's popularity.