Trixie doesn't so much get in trouble for boasting as she does for lying, had she not told a fib about beating an Ursa Major there would had never been anything to prove her wrong.
Anvilicious: Whatever way the Aesop is taken, it seems Ponyville does not take kindly to boasting. Or lying.
Broken Base: While Trixie herself was well-received, the episode itself is polarizing, due to the reasons listed under Designated Villain. Some love the episode specifically because it introduced Trixie and just like it for its own merits, others hate it for the mentioned problems.
Designated Villain: Trixie. While she's arrogant and boastful, one could easily chalk that up to her being a magician, an occupation for which being a Large Ham is largely demanded. She humilates several of the Mane Six, but only because they take up her challenge and try to upstage her, making themselves targets; and as Rainbow Dash herself lampshades, she and Rarity aren't always the most humble of ponies anyway. Most infamously, Trixie had little to do with the ursa minor attack, it was Snips and Snails who brought it to Ponyville on their own, Trixie tried to fight it off but didn't have the power.
Draco in Leather Pants: Trixie, hand-in-hand with the above trope. It helps that her "villainy" doesn't rate much higher than Alpha Bitch levels of jerkishness, and the universe punishes her more harshly than it punishes some of the actual villains. Ask some members of the fanbase, and you'd think she was a misunderstood Nice Guy getting attacked by the Mane Six.
Ensemble Darkhorse: Outside Celestia and the Mane Six, Trixie's popularity is rivaled only by Luna and Derpy/Ditzy, to the point some have called for her to become a recurring character.
Fridge Brilliance: It's fortunate for the episode's aesop that Fluttershy wasn't around when Snips and Snails brought the Ursa Minor to town, seeing as how she's great with all animals except dragons.
Also, astronomy buffs might notice that the Ursa has a certain constellation on its flank...
Also, the Ursa Minor has a long tail, just like in the myth of the constellation.
This could also be seen as Foreshadowing for the reveal at the end...
Hilarious in Hindsight: In this episode, Twilight's genuine magic beats out Trixie's parlor tricks. Two seasons later, Twilight's parlor tricks beat out Trixie's genuine magic. She just cannot catch a break.
Rarity's disgust at having green hair. When Season 4 introduces the "friendship rainbow" which incorporates the colors of the Mane Six's coats, Rarity's white occupies the middle position in the spectrum where green would normally go.
Unintentionally Sympathetic: Trixie really doesn't do anything wrong at the start of the episode short of exaggerating her accomplishments for the sake of her show (not exactly uncommon for performers). The show seems to expect us to root for the Mane Six who are essentially heckling her. And then the ursa minor attacks and Trixie, clearly frightened of it, tries to fight it off and is forced to admit her boasts were hollow.