BTS became more famous in the year 2014 after they seemed to have stepped up their music. They were more known to deal with rap music, but it seemed they blended into the pop style of music. They started becoming more famous when they visited 'American Hustle Life,' a Korean reality show at Los Angeles where they were introduced to hip-hop culture. In 2015, they released an album that has the track 'I Need U,' and this track was a major hit that also made them really more famous.
They also gained more popularity when they were invited to perform at the K-pop music festival (KCONs) in Europe and the United States. BTS being so famous is not magic, it is just a matter of personal rehearsals and hard work. They take a lot of time rehearsing what they know how to do well, and it pays them off. Simple!