Applejack is rather a difficult character to get into. I had a hard time getting into her since she admittedly wasn't the focus on any episode or at least wasn't well followed from a narrative level.
The reason why I became interested in Applejack was when I heard things like Applejack is Best Background Pony. And to that, I decided to rewatch S1 and S2 before the start of the summer this year (Season 2 ended). And at first, I liked her a lot but mostly 'cause of her accent and underdog status. But I just watched the series again recently and if you look hard enough, there are plenty of Applejack moments that stand out and make her character all the more admirable.
On the rough surface, people will complain that she has no episode from her narrative. They will say that she's overshadowed by her friends or even Background ponies (Derpy). And some will say she's uninteresting 'cause she doesn't have any flaws. And to that, I say: Some of that is true. All you have to do is look closer.
Applejack never had a episode from her perspective. It has always been from the perspectives of other ponies looking at Applejack. And y'know, as much as people complained about S2 Applejack ("AJ is best background pony cuz she had little episodes), i think the Last Roundup really showed how much she cared about the town. If it was RD, she'd care more about her reputation that she failed to get the money than actually feeling depressed that she cant get the money to help everypony.
A common recurring theme is that her friends or other characters steal her spotlight. Pinkie Pie and Derpy with the Last Roundup for example. But you must understand, it's because of the narrative the writers have chosen. As stated, all of her episodes always come off as the audience looking through the eyes of her friends and never through Applejack. This is true for Applebuck Season, where we dont follow Applejack 'cause she's... basically her brain not work right. So we instead, look through Pinkie Pie, Rainbow Dash, Fluttershy, and Twilight's perspective. Now, I can understand the whole 'Background Pony' thing but really, it's more than that. Personally, both narratives can provide a strong story about the character. There are tons of movies where we love the characters but aren't the main focus (Han Solo from New Hope). In the Last Round Up, the story is told in a very brilliant way (Promising beginning, Applejack missing, Applejack refuses to talk, chase scene, Applejack confesses and her friends appreciate her hard work nonetheless) and you learn from the perspective of the other characters about Applejack. I believe that she comes off as a stronger character because the plot unravels and you find out why she did this and what she went through. This is definitely one of the best Applejack episodes, and it should be commended too since it wasn't from her perspective unlike many of the other Character-favorite episodes people have (Lesson Zero, Sonic Rainboom, Hurricane Fluttershy, etc.).
Regarding her flaws, she definitely has the least flaws so far. She's definitely stubborn, refusing to back down most of the time and have it her way (Applebuck Season, Super Squeezy Cider 6000, etc.). She's also competitive, but it's healthier than Rainbow Dash's. In Fall Weather Friends, AJ is quicker to back down from her ego whereas Rainbow Dash takes things way too seriously and is the first to cheat. Remember, Applejack is the one who said: 'Remember, Rainbow, this is just a game.' And I think that says a lot about Applejack, how more mature and friendly she is. She also overworks herself twice but for different reasons. In Applebuck Season, she mostly overworked to prove her brother and others wrong. In the Last Roundup, she forced herself to do menial and embarassing tasks so she can return to bring joy to the town. I think the fact that she has the least flaws is a good thing and a bad thing. Bad because it seems like a lot of people tend to like characters with (glaring) flaws, and I agree with that generally. It's good because it shows how much more Applejack mature she is than the others and she's a good role model because of that.
And this brings me to the reason why I and others love Applejack. Applejack has proven time and time again to put herself in front of others because she cares deeply about them. She will step in as leader when Twilight isn't around (Friendship is Magic, Canterlot Wedding). She constantly saves her friends from harm (Friendship is Magic, Feeling Pinkie Keen, Dragonshy, etc.). She will sacrifice her chance to help herself and her family so her friends can achieve their dreams (Ticket Master). She volunteers to carry Fluttershy up the mountain when no one else would (Dragonshy). She does embarassing work but only because she cares about the ponies back town (Last Roundup). She cares about her friends doing honest work (Fallweather Friends, Super Squeezy Cider 6000). She welcomed Sweetie Belle (Sisterhooves Social) to participate in a race when Sweetie Belle wasn't her sister, and she sacrificed her chance at winning by switching places with Rarity so the two could get back together.
Yes there's other things like her country accent, her underdog status that appeal to some, and so on that make her so loved but above all, Applejack puts herself in front of others and her caring nature, is what makes her my- and probably others, our favourite pony. And when you realize it, it makes her all the more rewarding as a character.
PS I may come off as bashing Rainbow Dash but really, I love Rainbow Dash as much as I love the others... with Applejack being slightly higher :D I know Twilight is the main star but Rainbow Dash and Applejack just have more of a feud. It's also no surprise that Rainbow Dash is often overblown by the fans and media. For each merchandise Equestria Daily posts on their site, it's more than likely that Rainbow Dash, Pinkie Pie, and/or Twilight appear on them. Applejack rarely gets any face time on merch. But again, I only refer to Rainbow Dash because she and Applejack have a thing with one another.