Others have explained to you how and why Burke would have been involved in Bannon's invitation:- the Breitbart connection; and the fact that Burke was chair of the Institute which invited Bannon to speak. You don't find these things to be even "hints", never mind "proofs"?
"Media coverage" (whether in the US or elsewhere) is usually a mixture of reporting facts and then discussing their likely significance, as you know well. Often, it is the case that not all the facts are known (e.g., some details may be kept private), and in this case journalists will report such facts as are known, and then try to make sense of them. Of course, some journalists and commenters are less ethical, and they will also invent (or spin) 'facts'.   
Here, people are suggesting that Burke would have been involved, for the reasons stated, in inviting Bannon to talk to the 2014 conference. It is not a 'fact', but it is indeed very likely. If you have another explanation, please let us know it.