Awesome Music: The music and sound effects in the cartoon were all taken from the original Zelda game, including the iconic "Overworld" theme.
Base-Breaking Character: The show's incarnation of Zelda. On one hand she's recurrently more competent then the games' incarnation at the time. On the other hand, her attitude is only nominally better than Link's, and she still has enough Damsel in Distress moments to be viewed as an Ungrateful Bitch.
Engaging Chevrons: Ganon is guilty of this, providing an almost constant running narrative about his thoughts and Evil Plans regardless of whether anyone is listening. Link is almost as big an offender; the very first episode, "The Ringer," opens with him complaining to his reflection about how life in Hyrule is boring.
Hilarious in Hindsight: For a cartoon from the 80's, it sure ended up featuring a lot of things that would be prevalent in later games.
In the opening and a few episodes, Link rides horseback. His video game incarnations would start doing the same thing in Ocarina of Time.
Also related to Ocarina of Time, the episode "Fairies in the Spring" has Link and Zelda dealing with monsters made from water. Guess what Link ends up fighting in the Water Temple?
The episode in which Link exists only in spirit form and Zelda needs to help him is reversed in The Legend of Zelda: Spirit Tracks, with Zelda as the ghost and Link as the person to protect her.
Chances are many a fan have thought of Link's catchphrase of "Well Excuuuuuuuuuuuuse me, princess!" after witnessing his canon interactions with Princess Ruto, Tetra, and Midna.
In the beginning of the 5th episode, the King of Hyrule takes a nightly stroll along his castle, exclaiming "Nothing better after a good meal." In a later moment when Link is trying to swing in Zelda's bedroom after the King gives him some flowers for help, he says "Good luck, my boy!"
Some scene transitions to Ganon's throne room started with a shot of a smoking volcano such as this one that represented Death Mountain, long before it became the token fire/volcano dungeon in Ocarina of Time and subsequent games.
Link shield-surfing became possible in The Legend of Zelda: Breath of the Wild. Furthermore, the cartoon's premise, in which Link serves as Zelda's personal bodyguard at a time when Hyrule is relatively at peace, resembles the pre-Calamity backstory of the two characters in Breath of the Wild.
Ho Yay: Link gets a kiss from the king in one episode.
Spryte. Say what you will on Navi or Fi, they at least served a purpose in Link's journey, even it's really for gameplay purposes. Spryte serves no purpose other than to be an annoying pest with an unrequited crush on Link who keeps interfering with his attempts to kiss Zelda.
Link is not safe either. A lot of fans find his constant complaining, his catchphrase, his tendency to be The Load in a series he's supposed to be the protagonist of rather annoying. Also, he seems to care more about getting a kiss from Zelda rather than helping others.
Vindicated by History: Though the show does have its detractors (especially from fans of the games), the Zelda cartoon is considered one of the best game-to-show adaptations for its time, citing the impressive action sequences and expansions to the lore of the first two games. In addition, as the "Hilarious in Hindsight" entries would tell you, it had a lot of elements that would be seen in future games in the franchise. As for the character portrayals (Link in particular), it's easy to forget that at the time, there was no real standard set when it came to stories in the franchise. A Link to the Past and Ocarina of Time weren't even made yet; all the story-based elements were All There in the Manual, and even then it was still mostly fairly simple. As a major example of this trope, James Rolfe himself remembers the show quite fondly.