The break down is this: in tennis, like in many other sports, there is a code of conduct. I am not sure of the exact terminology, but there are minor infractions and major ones. Major ones make you lose the match instantly. This happens from time to time, usually it's when a player throw his or her racket and it hits an official, a ball kid or a spectator. Nalbandian actually lost the final in Queen's against, I beleive, Cilic a few years back, because he kicked one of these little advertisement panel that are in front of the line judges and it hit the judge in the leg, injuring him! Now that was dumb! Anyway, minor infractions are things like swearing, breaking your racket, etc. At the first minor infraction you get a warning, at the second one you get a point penalty, at the third you get a game penalty and I believe the fourth is a set penalty and that at the fifth you lose the match, but I have never witnessed it. Warnings are pretty common, point penalties happen from time to time, usually in long men's matches because the guy forgot he already got a warning and sent a ball flying off the stadium in anger or something... game penalties are really rare. Also it should be noted that the warning you get for taking too much time between points when serving is another beast (though I am not sure about the warning for taking too much time when receiving, but that almost never happens so yeah).
In today's match Serena Williams got a warning early in the 2nd set for a minor offense: coaching. In grand slam tennis you see, there is no coaching. The people in the box can cheer, say well played, but they can't say "Slice more!", "Play her backhand!" or whatever. One of the interesting thing in tennis is that, like in chess, the athlete is alone during the match and has to figure a strategy or adjust the one that has been planned with the coach. It's a rule that is notoriously poorly and unevenly applied. In part, because the refs often don't understand or plainly can't hear what the coaches are saying. Tony Nadal spent the whole match talking to his nephew earlier in his career, Djokovic and his then coach Becker all but admitted that they used code signs during matches, etc. In today's match, Serena Williams' coach, Patrick Mouratoglou was dumb enough to use pretty clear hand signs to convey to his charge that should should play toward the middle of the court to let Osaka create her own angles. The chair umpire Carlos Ramos saw that and gave a warning to Serena Williams.
Williams was more than a little displeased. Not only did she not see her coach's gestures (which btw is irrelevant), but she felt she was accused of cheating which was unbearable to her (she uttered a quote that is destined to become famous: "I prefer to lose than to win by cheating"). Tension was high (also helped by the fact Osaka had won the first set by a resounding 6-2) and some chatting ensured during the change of ends. This anger actually helped Williams as she broke Osaka in the next game, but the resourceful and imperturbable japanese player broke back immediately. This caused Williams to shatter her racket.
The umpire has some leeway to decide what amount to a racket abuse violation. Dropping it is ok. Throwing it is fine unless you do it over and over (players sometime get a soft warning... they are warned that they will get a warning if they go on). Breaking the racket on purpose though is an automatic minor infraction and a second minor infraction is an automatic point penalty. This did nothing to better William's mood. Additional chatter ensued (I think Ramos being somewhat sympathetic to her earlier plight wrongly gave her the impression that the warning was voided?) and when play resumed, Osaka held to love and then proceeded to break Williams again to get ahead 4-3.
At the change of ends, a furious Serena Williams just tongue lashed the umpire during a minute. Demanding apologies, calling him a thief and also a liar I believe, vaguely threatening him that he made a career ending decision, etc. That was a little too much for the phlegmatic portuguese umpire who gave another minor infraction to Williams. Third minor infraction: game penalty. Obviously Serena Williams was not too happy about it to put it mildly, but a call to the supervisor (someone who can correct the umpire) confirmed the decision. Feeling robbed, the 9th time finalist and 6 times champion held her serve convincingly, pushed by a sympathetic crowd. For a moment it felt like the drama could be a turning point in the match, boosting the veteran and destabilizing the rookie. In some alternate universe, Williams broke back a shaking Osaka, clutched the 2nd set and steamrolled her way to victory in the third. In our universe, a cold blooded Osaka played a great rally, fired an ace and two service winners to win her first Us Open title.
I guess you can argue that the initial warning for coaching was a bit harsh considering they are almost never called. You can also argue that Ramos might have used a little more psychology and give Williams a soft warning after the point penalty, reminding her she has to watch her tone and tongue as a third infraction would force him to give her a game penalty (maybe he did not because earlier in the week another umpire, Mohamed Lahyani, got side-lined because he used a little too much psychology with Kyrgios, almost giving him a Davis Cup style pep-talk at a change of ends!). That is fair, but on the other hand, a veteran professional player should be able to understand the situation she is in and not break a racket 5 min after getting a warning, know she better focus on the match rather than on decisions she cannot change and realize nothing good can come out of accusing the umpire of being a thief. Still, some Williams fans are dejected (former player Marion Bartoli who did the commenting on the channel I watched certainly was!). In the end though, the point and game penalty came on Osaka's serve so it is very likely it had no impact on the outcome considering how well she played through the whole final. If anything, feeling slighted fueled Williams as she was heading for an unceremonious 6-2 6-2 loss without it.