It's quite simple, and it's called tact. Most athletes do a terrible job in describing slights from their perspective. Fair or not, words matter. The ability to articulate an idea, at times is more persuasive than the actual idea itself, hence descriptors such as "slick-talker". Most athletes due to the nature of their occupation (not lack of intellect), can't match politicians and shrewd businessmen in eloquence. That is to be expected but does not excuse them from grabbing a mic and saying anything, then being upset with negative perception. You're in a industry that relies heavily on consumer perception and consumption. Remember that.
I wrote this after reading an article about Buffalo Bills Defensive Tackle Marcell Dareus. It's a perfect example of how not to express contract dissatisfaction. Let me provide a bit of context, he's a very good player, and the Bills offered him a contract exceeding $90 million over 6 years. These are some of his quotes:
"I feel like they don't want me here as bad as I want to be here, as bad as the fans and my team wants me here"
"I feel like they're saying, 'Whatever. You come a dime a dozen."
Now it's not my place, to say whether he should feel slighted about his contract situation and the legitimacy of it, but how he described his perspective was terrible. In most situations it's probably a good idea for athletes to publicly stay away from contract talks, but if you decide that you must, then come prepared. Make some great points of why you feel slighted, why your asking price is equitable. Don't just say the team does not want you there or is treating you like you come a dime a dozen after they offered over $90 million. That is not an average NFL Salary, or even close to one. It comes off as overly arrogant, entitled and further distancing the athletes from fans. That's important because despite the fact that owners are much wealthier than players, fans feel closer to the owners than to athletes.
Professional team sports have a complicated market (please do not use the term "free market"), so it's not absurd that an athlete wouldn't be satisfied with $15 million a year if he thinks he's worth closer to $20 million. Regardless of whether that is accurate or not is not the point. The point is that if you want public perception to favor you, you have to speak in a manner that the public can relate to. We're in a ridiculous tough job-market where everyday more people are getting paid less to do more. It's not any athletes fault, but when you say you're being treated like you come a dime a dozen over $90 million, people won't be able to see your perspective at all. Often times it's not really that people are favoring management, just that they are choosing not to favor athletes that come across as (for lack of a better word) spoiled.