Here's what happened.
Brady is anal about the football. He knows JJ but only knows McNally tangentially, by a nickname, "Burt" (he doesn't even know him well enough to know that his nickname is "Bird", not "Burt"). McN and JJ make sure the balls are prepared as Brady likes, because if they don't, he can be a real pain in the ass about it. It's his livelihood, after all. So they bust his balls behind his back, like most employees do to their bosses.
The league, however, is NOT anal about the footballs. They just don't care. When the Chargers illegally put stickum on the balls, they get fined $20k, but not for tampering with the footballs - rather, for not cooperating with the investigation. When the Panthers and Vikings tamper with the balls during a game, all they get is a warning. There is no real tight procedure or chain of custody vis-a-vis the footballs. They don't record measurements. Everything is completely inexact. Mostly because, in the entire history of the NFL, nobody has ever cared about it that much. Nobody even thinks that the regulations will be in jeopardy on really cold days when the laws of physics force footballs under the legal psi. None of the refs ever have even heard of the Ideal Gas Law or consider its impact on footballs. Nobody cares ever.
One day the Pats get footballs back from the refs and the footballs are at 16psi. Brady can't feel the difference of 0.3 psi, but he sure can tell when a ball that he wants at 12.5 is 3.5 psi higher than that. Either the refs deliberately over inflate the balls, which would open up a whole other can of worms, or they just do the "inflate, squeeze, put them in the bag" routine. No precision. Again, because nobody cares.
Except that's too much imprecision and Brady lets the ball guys have it. Come on guys, how can you let me play with footballs like that? Smarten up. 12.5!! He is crystal clear about 12.5.
Fast forward to the AFC Divisional game. The Pats come back and beat the Ravens. Harbaugh is embarrassed and pissed especially because of the formation stuff (which was perfectly legal). Brady says afterward that he should know the rulebook better. This further embarrasses and pisses Harbaugh off. So he tips off the Colts to the idea that the Pats screw with the footballs. The Colts alert the NFL, which is filled with people like Kensil who hate the Pats.
That information is passed on to Anderson, and they make sure to check the balls before the game. But, because nobody cares about it, all he does is a very imprecise check using the logo gauge, and doesn't record them. Which, if he was part of a "sting", he sure would have been more precise. But he's the ref. He was told to make sure to measure the balls, so he does. Then McN gets the balls and waits for the NFC title game to finish. It does. He gets up, walks past dozens of NFL officials with the bag of balls, without anyone saying anything. Why? Because they don't frigging care. If anyone does care, they sure aren't showing it.
He stops in the restroom to go to the bathroom because the NFC title game took extra long and he has to go. He's in there long enough to take a leak. He brings the balls in there because they're his responsibility and he doesn't want to let them out of his sight. He then travels to the spot on the field where he's supposed to be.
Anderson, meanwhile, shows up on the field and finds McN right where he's supposed to be. Doesn't chew him out. Nothing. Just…normal. If Anderson is so distraught at losing track of Anderson, and what that might mean for the integrity of the game, all he has to do is replace McN's footballs with the backup balls and have the Pats' balls checked and (if necessary) re-inflated. He doesn't do any of this. Why? Because it's totally insignificant.
The game begins and then Jackson intercepts Brady. The ball is brought to the Colts' sideline. One of the Colts' people, trying to catch the Pats in this scheme, tests the ball, in violation of NFL rules (he shouldn't be tampering with the ball at all). Lo and behold, thanks to the Ideal Gas Law, it's under 12.5. He alerts Kensil. Halftime comes. Kensil tells the Pats' ball guys that they're F**ed, because their footballs are under inflated.
The balls are brought in at halftime and Anderson and company measure them again. Again, very little precision, and nothing recorded to compare them to. They switch gauges and it's just silly. But they record the Pats' footballs and then just do 4 Colts' balls. Almost all of the Pats' footballs and 3 of the 4 Colts' footballs are under the legal psi. The Pats', moreso, so it looks bad (and this is where people get the "If it's science, why didn't the Colts' footballs deflate too" - not knowing that the Colts' footballs started off at a higher psi to begin with). They have plenty of time to do all the Colts' balls, but they don't. Why? Who knows. The Pats' balls are reinflated but the Colts' are not, which means that the Colts played the entire second half with even more deflated footballs.
They start the inquisition immediately after the game by talking to McN. The next morning Brady goes on D&C and they spring this news on him and he is like, "Wow, now I've heard it all." Immediately after that interview he calls JJ and is like WTF is going on? That leads to several more calls and texts over the next few days as Brady checks in to see how he's doing. After all, this is craziness.
Meanwhile, someone from the NFL office leaks to Mortensen that 11 of the 12 Pats' balls were at least 2 psi under the limit, suggesting something outside of a scientific explanation. This is something nefarious. Immediately that report spreads like wildfire and the narrative is created - the Pats were deflating footballs to gain a competitive advantage. This despite the fact that Brady played much better in the second half with legally inflated footballs.
The narrative is set. The NFL, knowing the actual measurements (and that Mortensen's report is wrong), and knowing that they are explained by science, have a chance to correct the Mort tweet and bring this story to a place of neutrality. They don't. They let this narrative play out. One short statement from the NFL saying that Mortensen is wrong, and here are the measurements (of the Pats' and Colts' balls), would have been sufficient to at least bring the story back to a neutral place. But no. They let it play out and the integrity of Belichick, the Patriots, and most importantly, Tom Brady, is put into question. I say most importantly Brady because BB and the Pats have long been considered "cheaters" due to Spygate, but Brady has never had his integrity questioned ever. He is, in many ways, the face of the NFL.
In the 2 weeks leading up to the Super Bowl, the Patriots are left twisting in the wind as the NFL allows this narrative, based on an erroneous tweet by Mortensen that the NFL *KNOWS IS WRONG*, to fester.
As public pressure builds on the Patriots and on the league to get them, they happily comply. Wells is brought in. An "independent" investigation begins - and it's anything but "independent". The Pats cooperate. Brady cooperates. Science experiments back the Patriots' explanation. Wells hires a firm that argued that smoking doesn't contribute to lung cancer to be their science experts. As Wells publishes his report, he leaves out anything that supports the Patriots, or is even neutral. He includes everything that even hints of Patriots' wrongdoing. He makes assumptions and derives conclusions from those assumptions.
His conclusion, even after all that, is not that the Patriots definitely did something wrong. BB, Kraft, and the organization are exonerated. McN and JJ are thought "more probably than not" to have deflated footballs. Brady is concluded to "more probably than not" have had "general awareness" of McN and JJ's actions. This is the strongest claim of guilt that Wells can muster, after 100+ days and 243 pages worth of work.
The report is passed on to Goodell and Vincent. The Patriots are not spoken with before hand to give them an opportunity to respond. Goodell and Vincent put their finger in the air to test the winds, and the winds are overwhelmingly against the Patriots.
They then levy an unbelievably harsh penalty. Four games (and $2 million) for Brady, $1 million for the team, and a first and fourth round draft picks. This despite the fact that the original "crime" warrants a $25k penalty per the NFL rulebook. They are cited for not cooperating (which turns out to not be true), and for prior transgressions (Spygate). This despite the fact that other teams are multiple time rule breakers (the Jets, with three violations since 2010, and no extra penalties levied on them) do not receive the same treatment. The punishment for the "crime" of not cooperating is completely out of step with similar infractions (the Chargers received a $20k fine for not cooperating; Brett Favre in 2010 was fined $50k for not cooperating).
So the penalty fits neither the original "crime" ($25k), the lack of cooperation (see Favre and SD), nor the idea of multiple-time offenders (see the Jets). And the penalty is all based on something that is easily explained by science, or at most, is hardly demonstrated by the known facts.
How did this all happen? It seems clear that someone was out to get the Patriots. Harbaugh had motive - he was embarrassed by them and they are his primary competition in the AFC). The Colts had motive (rivals). Kensil had motive (former Jets guy, hates the Patriots). So many key players involved in this drama had motive to see the Pats take a big hit. And so they initiated this narrative and let it play out when they could have stopped it right in its tracks, and then they commissioned a one-sided report that ignored all the information that would have put the Patriots even in a neutral, let alone positive, light, and instead only presented any evidence (along with their accompanying commentary) that put the Patriots in a negative light.
In other words, this whole thing could have been prevented from the very beginning. But the NFL let it go and let it grow and let it fester and let it explode. What was originally something NOBODY cared about EVER in the history of the NFL became one of the biggest scandals in the sport's history. And, as it turns out, it's all the NFL's fault.