The nickname "The Big Three" came about due to their immense worldwide popularity and length. With that, they were always featured in front of the shared Shonen Jump covers and always had bigger images for their main protagonists: Luffy, Naruto and Ichigo.
While the three series weren't the biggest sellers in the magazine, Hunter x Hunter sold more per volume than Bleach for example, they kept their position on the covers due to their overall popularity.
These three manga have each received long running anime series, Bleach has 366 episodes, Naruto (+Shippuden) has 720 episodes, and One Piece has 979 episodes as of June 2021, with One Piece and Boruto, Naruto Shippuden's sequel, still airing. These statistics do not include the spin-offs, sequels, and movies, so the full numbers for entire franchises are higher.
All three are shonen series, which usually focus on topics such as adventure, growth, and action. This genre normally has a lot of subplots and one main, overarching plot. Fighting is also a main part of most shonen series: the protagonists almost always have a superpower and are very muscle headed in nature.
The term is commonly used in various otaku discussions, and The Big Three tend to attract similar audience so it's likely that fans of one series will at least try to check out the rest.
Sometimes fans discuss what the Big Three of other genres could be, or try to find a New Big Three for shonen. During Bleach's decline in sales and popularity, Toriko rose to popularity and started to become featured in front of Bleach on the shared Jump covers, effectively taking its place in the big three. Soon after Toriko & Bleach started to lose popularity and eventually end alongside Naruto, other series became featured more often in front with One Piece, including - My Hero Academia, Haikyuu, Assassination Classroom, and recently Kimetsu no Yaiba, but its unlikely that there will ever be other three simultaneously running series that would be able to challenge the original big three in popularity.
During the Golden Age of Weekly Shonen Jump, when its circulation was at its peak, according to fans a "Big 3" equivalent for early 90s would be Dragon Ball, Slam Dunk and Yu Yu Hakusho. While late 90s equivalent would be One Piece, Rurouni Kenshin and Hunter x Hunter.