To set this up, I'll say that I actually liked all of the rest of season 9, it's only the last episode that I hate. And, to be clear, I haven't read any of the other criticism of the finale. Literally none of it. This is all based solely on my own viewing of the series and the last episode.
So the basic reason I hated it was that I think that the last episode betrayed basically everything that the show had built up to that point. Here, in detail, is how I think they did that.
As the show's lead, Ted is who the show is really about. In the end, he loses the love of his life, skipped out on the job of his life, and any growth he had experienced during the show's nine years is eliminated when he goes back to being the same person he was at the end. The show is supposed to be a comedy, but for Ted is one long series of tragedies that, in the end, don't lead him to something better, other than to hold it and have it taken away.
Robin shows no growth, either, and is not portrayed as particularly rewarded for her life choices. By the end, while her career worked out great, she was shown as not being able to handle a marriage when a guy who fit with her perfectly fine, giving up on that after three years, and she's left with the final choice of becoming the old woman with all the pets, or get back together with a guy who she's not particularly compatible with, who she tried relationships with more than once and they didn't work out, and who, it's obvious isn't much different than the guy she turned down in the beginning of the series and more than once after that. Despite being a beautiful, intelligent, and successful woman, the only man she can attract at the end of the series is her old boyfriend. And she's happy about it, despite knowing that she'll never be the love of Ted's life. He's already had that and she'll never be able to live up to Ted's memory of his deceased wife, who he very clearly loved more than anyone he ever dated. The show is saying that her best option is to be with the guy with whom she'll always be second best. Or maybe even third best, if his children are included. Nothing about this ending is good for Robin or says anything positive about women in general (particularly combined with the Barney betrayal).
Ted & Robin as a couple
There's little to no chance their relationship will last very long. If Robin couldn't make it with Barney, a guy perfectly suited for her and who changed his person to become a better man to make her happy, how's she going to make it with Ted, whom she has little in common with and with whom, after a while, she grows more annoyed with based on his inherent personality (and who already had the love of his life, to whom she can never live up to). They were great as friends, but they lacked the spark to go beyond that. How is it going to now appear after she married and divorced his best friend and he lost the love of his life. They haven't changed, really much at all, since they previously dated, except that both are a little more unhappy and dissatisfied with life. It's hard to see how they work things out in the long run.
Barney is another character whose growth over the years, which was one of the key subplots of the series, is erased. In his case, after he can't make a marriage work with the one woman he loved enough, and he searched high and low to find, to change his very being for, he reverts back to being as bad or worse than he was before, hoping to capitalize on young women's "daddy issues" because he now is as old as their daddies. Supposedly he changes again because of his daughter, but is that really a sign of growth, or is that him just trying to possess and control yet another woman? Has he grown or has he just transferred his feelings that women are only props for him from his dates to his daughter. The kicker is when he slut-shames two young women in a bar for dressing the way the women he exploited for years dressed. Women aren't people to him, they're only valid if they live up to his standards of what a woman should be. Barney, in fact, regresses in the finale. Notably, we don't learn who the mother of Barney's daughter is, because she means nothing to Barney.
Robin & Barney as a couple
After investing the viewer in this couple, which shouldn't have happened, for years, the show throws them away in mere seconds over what appears to be no significant issue. For years, we are made to believe they should be together, and we eventually sign on and we're happy to see their wedding day finally come. Then we are given no real reason why they end up apart and given no real time to understand or mourn their relationship, which the entire last season was about in one way or another. They told us that this couple grew in recent years so they could be a mature couple that can love each other, and then it's all gone in a meaningless poof.
Barney & Ted's friendship & the Bro Code
We're taught to like Barney early on, despite his misogyny, by being shown that he is a good guy at heart and by knowing that he'll never go too far because he lives his life by a strict "moral code," the Bro Code. But most assuredly, one of the key components of any bro code is "don't date your bro's ex-girlfriend." Barney breaks this one in a big, dishonest way. But we're taught by the show that it's okay, and Ted learns to accept it, because Barney has grown up and because he and Robin are made for each other in a way that Ted and Robin weren't. It's the only way that such a code violation could possibly be okay. But Barney and Robin end their marriage after three years over minor problems, and they never try again, effectively meaning that Barney screwed Ted over for no particular reason. And in the later parts of the episode, in the future, it is clear that Barney and Ted drift apart and don't really talk much. If don't date your friend's ex-girlfriend is part of the Bro Code, which it has to be, don't date your friend's ex-wife has to be an even bigger rule. And Ted jumps right back in there, either not caring about Barney or with an intention of getting back at Barney for Barney's violation of the code.
Lily and Marshall
The one couple that is sane and healthy and grows into happiness is kind of tossed aside towards the end of the episode, which suggests that the show wasn't about them much at all, it was about the love triangle. Lily and Marshall, despite being integral to the series, aren't integral to its finale.
Rarely has a show's titular character been given such short shrift. She doesn't even appear for the vast bulk of the series, so we're left wondering about her for years. Then they finally introduce us to her and she actually lives up to the 8 years of hype. She is Ted's perfect woman. And she's realistic. She's not a stereotype. And then she's dead, before we even really get to know her. And it turns out that she's nothing more than a prop for the male lead. She's his Holy Grail for 9 years, he gets her and she's all that he wants, then she's haphazardly tossed aside and Ted goes back to his ex-girlfriend who never really made sense for him anyway. At best, they're saying "she allowed Ted to grow enough that he could finally be right for Robin," making her a prop. At worst, they're saying she's meaningless.
Basic story structure
The show was very explicitly, for years, structurally focused on the end point being, you know, "How I Met Your Mother." From the flashback story structure with the future narration, to the final season spacing out how each character met the mother, the inevitable conclusion, structure-wise, would be to end the show on the meeting of Ted and the mother. And they did that. And tacked on another ending that undercut that previous structure. This is not a good story structure and it diffuses whatever you're trying to do with the narrative. Beyond that, the show is clearly an absurdist fantasy comedy. Such a show demands a happy ending. That's why we watch such shows. We got that happy ending, and then it was casually erased.
The idea of human growth
To sum up, the show is about the growth of the three main characters for nine seasons, then all that growth is erased in the last half hour. And the only couple shown to actually grow is tossed aside to focus on the people whose growth is erased.
Barney and Robin have it, but they throw it away. Ted has it, but it's taken from him. Lily and Marshall have it, but they are incidental in the end. Happiness isn't allowed in the finale.
For years, Bob Saget did the future narration with no screen time. And it turns out to be the best, most respectable work he's ever done. In the one chance he would have to show his face on the screen, they show Josh Radnour in make-up. If they were going to do that, why have Saget do the voice-over in the first place. Saget loses a shot to be a real person in the HIMYM universe, despite deserving the screen time.
Hope and perseverance
If there were two themes that were inherent to the show and to Ted's life, they were hope and perseverance. Ted goes through a lot of trials and tribulations in his search for love. And he loses. A lot. But we're kept interested because we're promised that the search will eventually pay off. It's right there in the title, he will meet the mother and so he should remain hopeful and persevere. And so should we, because the viewers are Ted. Our lives, even if they don't have the insane number of bumps that Ted's do, will be okay. Eventually, if we keep at it, we will win, so we should never give up hope. The finale says "screw that, life sucks, and Ted loses and you probably will, too."
And I take that last one personally. I started watching the show during a period when those kind of roadbumps had pushed me to the edge and I was on the verge of giving up. But I quickly realized that Ted was me (in some way, each of the characters represented a part of me or a phase I went through), and the show helped me regain hope that I could grow into a more mature person and I would find the person who I could find happiness with. If Ted could do it, with all the obstacles he faced, so could I. And what the finale says is that Ted might get a glimpse of that happiness, but it's going to be taken away because he can't have it (or doesn't deserve it) and there is a pretty good chance the same is true for me, the surrogate Ted. Sure, the writers weren't writing specifically to me, but they were writing to the many Teds in the world, and this ending says, that hope and perseverance are myths and we should just probably settle for something less than we want or deserve, because we can't have it. That's not why us Teds of the world watch shows like HIMYM.