What sold me on the lore of League initially was the League itself.
I realize it's not canon anymore, but when I first joined in, there was this sense that the entire playerbase was a part of the League's lore. Obviously not all of the millions of players could be canon summoners, given how rare and powerful they supposedly were, but I wasn't going to meet all of the millions of players, so that was fine. I don't remember who it was but when I bought my first champ with Influence Points, I remember constructing this mini-scene in my head of meeting with the champion personally and brokering a deal with them on what kind of matches I could and couldn't summon them in. At the time I was more into Dota because the characters in that game spoke more during the match and after kills, which gave you a better sense of who they are. (the more recent champions do this now, but when I was first playing the rift felt almost eerily silent) In my mind, Dota had the more active characters but League's major selling point in its lore was the sense that you, the player, were involved in shaping Runeterra as a summoner. (Remember when Noxus vs Ionia was solved via RL game results?) Hell, Sona talks directly to her summoner.
Sure, there was s#@$ like 'Lollipoppy' that Riot clearly wasn't taking seriously, but w/e, seriousness and maturity are overrated anyway; it's a game.
To be clear, I get why the League was cut. As cool as it was, it was also extremely limiting for the lore. Shurima's champions as they exist now would not be possible in the old lore. However, it was still sad to see the game's very namesake just... no longer exist, and it threw the lore into chaos. At the time I was more of a Lore guy than a gameplay guy when it came to League so I just kinda... drifted off, for a long while. I came in and out, different champions got released or revamped...
What sold me on the new lore was (as you might have guessed) the rework to Shurima's lore; specifically Xerath's story. Driven by his friend by what he saw as betrayal, only to then realize 'hey he wasn't', and then betray him anyway because he'd sacrificed too much to abandon his plans now. It was great, and tied the Shuriman champions together in a way that I thought was amazing. Then the Harrowing happened.
A lot of stories and universes promise that "nothing will ever be the same" after a particular event, but for fighting games, comics and MOBAs I've noticed that the status quo is usually god, at least to some extent. Nothing sticks. Before the Harrowing I figured that Nasus and Renekton would never actually fight, because then one of them would die and be removed from the Rift. Then Gangplank got blown up and had his entire kit reworked.
Yeah, sure, Gangplank didn't DIE, but he was definitely changed by his encounter with MF, right?
The moment I was sold on this game and started following it more than casually was the same moment I realized League's lore is ONGOING; Characters develop, grow and change, and in ways that actually matter. When Taliyah came along and we saw Yasuo not only did it help them both grow (in Yasuo's case in a way I didn't expect at all) it looked almost like Yas was being introduced to new readers, which was a great plan! Camille's dropping hints at Viktor showing up in her lore, and is implied to have made Vi an orphan in her voice lines. That opens up a lot of possibilities, and now I'm sure they'll actually be expanded upon rather than just introduced and left in the air. (coughRivenProbablyIsWhoKilledYasuo'sElderButIsAlsoProbablyNotTheKindaPersonHe'sExpectingcough)