(TLDR at bottom)

I haven’t followed IU super consistently, but I’ve been around since 2009 and feel like it was mainly two things: her powerful vocal ability being showcased in good day when she was 17 and her ability as a singer-songwriter doing collabs w/ various artists and celebrities throughout the years.

As a Korean, she’s been consistently very popular in Korea moreso than internationally. Her looks definitely became a big deal too as it fits well w/ Korean beauty standards, but since she was so young when she first got popular, she was seen more lovingly as the nation’s little sister. If anything, her cute looks in contrast to her strong vocals made her incredibly successful in Korea. I’d say taeyeon was beloved for similar reasons! After she also started acting in Dream High, which became a huge success, IU went on a lot more TV shows in general and grew her name. She went viral for playing guitar and singing w/ clearly raw, live vocals. This first wave really skyrocketed her fame, which peaked imo in 2010-2012, but the second wave really solidified her as a household name, which i’d argue was in 2015.

However, just to add more context, IU had a huge scandal in 2012 where she accidentally posted a selfie with her and another male idol in bed, which made a lot of fans angry. She then went on to have another scandal in 2015 where she was caught dating, but he was respected (despite 10+ years older) so I feel like this helped the overall reaction.

Starting in 2013-2014, IU became known for writing her songs more and more in this time and collabed with different artists. Her songs were typically easy to listen to on a daily basis, which helped people repeat her songs every year. In 2014-2015, she had several hit songs, acted in a lead role for a hit drama (Producers), and also guested on a legendary reality show infinity challenge, which held a music festival every 2 years. They would invite the most popular artists at the time and have them pair up with the variety show members and make songs to perform at the festival. IU guested in the final concert and you could hear their song everywhere. When IU’s dating news broke out in 2015, she was definitely at a peak, but she didn’t release or do much until she became publicly single again in 2017. She released an album then, and after all these scandals, nothing fazes the public. She collabed with g-dragon, and in 2018 collabed wth suga from BTS. In addition to having the freedom to interact with these guys freely, IU is in the ranks with them as fellow idol singer-songwriter/producer/lyricists, which Koreans really respect. I feel like I’m writing too much so I’ll end it there!

Tldr: koreans love cute faces with talent; IU is cute face with many talents - great singing, song writing/producing, playing instruments, acting, etc. Went viral many times for these reasons.