This is... not a favourite of mine, I'll be honest.
Boast Busters isn't wholly obnoxious, or entirely boring, or anything of that sort, but I don't really like it as an episode. It has a few funny moments, but they're less frequent than previous episodes and in between them there are moments that aren't really that fun to watch.
The new character introduced here, Trixie, is established a bit differently from Gilda. Gilda was made obviously mean by contrasting her behaviour with others. Trixie is made obviously mean by making her just plain unpleasant. Her screen presence borders on unbearable as she insults her audience and expresses her bloated self-image, and part of this falls upon her voice actor. The voice Trixie is given is obnoxiously self-important, with exaggerated syllables and a constant tone of arrogance.
Twilight somehow worries that this is how she'll come off if she so much as uses her magic to best Trixie. How she came to this conclusion is beyond me. This seems like a case where the storytelling is built mostly around the eventual lesson, and this comes at the expense of the story's quality. This internal conflict doesn't really show much development for Twilight in my opinion, though that can be forgiven as this is still early in the series and before it became character development-intensive.
Characterization in this episode is particularly flat otherwise. Characters mostly exist to react to Trixie, except Spike, who mostly exists to react to everyone in a slightly obnoxious manner. These characters are definitely not at their best (Rainbow Dash in particular), but at the very least some of them are entertaining enough. Rarity in particular puts on an enjoyable show.
Funny moments still occur, like the aforementioned Rarity scene, but a lot of the episode is either Trixie's grating personality or Twilight's mediocre conflict. Even the conflict resolution isn't especially satisfying, as Twilight basically lulls the bear to sleep and gives it a bottle. It's a rather limp climax for the episode.
The real ending, however, is one thing of note here. Trixie has, for once, had to bring herself down to the level of others, which may have showed her that boasting so aggressively isn't the right thing for her to do. She's been given things to think about, whereas Gilda probably just shrugged the ponies off as losers. Trixie might learn her lesson... and, in her next appearance, she did. I don't see why people liked Trixie so much from this episode, but I can sorta see why they might think she could be sympathetic.
With all of that said, I'll conclude that this episode is far from a high point of season 1. It's just not really an episode I enjoy a lot. Later episodes, after MLP became more ambitious, failed harder, but I remember this one for being the first I really don't like.