It was largely an extension of something that started during season 2: a certain segment of the fandom became incredibly disillusioned with any episode that didn't push the main plot forward, writing them off as filler. Even really good episodes like The Tales of Ba Sing Se provoked some controversy because they were perceived as delaying the resolution of ongoing storylines, like the question of Appa's fate. But Book 2 ended on a run of highly interconnected episodes, and people were expecting more of this with Book 3. But instead the pace screeched to a near halt after the premiere episode and we went back to something more like Book 1's episodic style. Now in retrospect, I really love a lot of the first half of Book 3 episodes. But at the time, people were really impatient to get to the invasion, and had less interest in more lighthearted episodes in particular.
The Beach in particular was really controversial, largely because the fandom was divided into competing camps over the issue of characters and shipping. Honestly, ATLA gave Harry Potter a run for its money for intensity and longevity of its shipping wars. Kataang and Zutara were the main ships, and the creators of the show had become aware of the shipping wars at some point during Book 2, so they did a lot of teasing about this during Book 3, which contributed to the crying foul after the end of the show.
A lot of this stuff is probably less noticeable when marathoning the entire show, but we had a lot of time to mull over all these little things because of the way the show aired. During the first two seasons, there would often be 2-3 weeks between episodes, and during the summer of 2006 this became more like 2-3 months. But there was a nine month gap between the end of Book 2 and the start of Book 3. Then, after the first 11 episodes, the show went off the air with almost no explanation, and stayed off for the next seven months, until they aired the final 10 episodes in the space of a week (although a couple had leaked online in the interval). This was obviously outside of the control of the creators of the show (I think it was somehow related to the writer's strike, but the timing might have been a coincidence), but a lot of fans didn't take it all that well.
Then of course the finale was controversial, as Laughing Banana alluded to. I don't recall the rock element being complained about as much then as it sometimes is now, but a lot of people (myself included, at the time) thought that the show pulled a huge cop out by not having Aang kill Ozai, or at least in the manner in which they did. I do still wish that they'd done something a little less out-of-the-blue than the lion turtle, and no, I don't think having a picture of a lion turtle in The Library counts as foreshadowing. But I know a lot of people disagreed at the time and loved the finale unreservedly, and many (probably even more) still do.
But I gotta stress that my perspective on Book 3 has really changed over time, probably in part because I've grown up a lot since that time (I was 13 then), and I think it has a lot of the best individual episodes of the show, although I don't think it comes together quite as well as Book 2 does. I think this is true for a lot of fans, because rewatching the show marathon-style makes a lot of the pacing issues moot, and helps highlight the strengths of the season. And I think most people moved on from the shipping wars. But I have encountered some people who still carry a grudge against Book 3 on ATLA fan forums. Which is kinda ironic, but that's a phenomenon you see in a lot of fandoms.
