It is a sad fact, about life on Earth, that there tends not to be the institutional capacity to support all students in developing their own unique talents to the fullest. While certainly some leeway can be granted, for people to elect to take classes in areas they find especially interesting, academies primarily aim for the least common denominator to be reasonably successful. A certain type of student—albeit, perhaps, an exceptional type—will find this fact incredibly frustrating.
Hope's Peak Academy is unique in standing against this trend. If you're admitted to Hope's Peak Academy, you will receive whatever resources you need to become the greatest version of yourself you could possibly be. Though Hope's Peak alumni walk many different paths in life, they're always the best of the best in whatever they've chosen to do. So while the school remains very small, as the level of individualized focus it gives isn't easily scalable, its name nonetheless carries a certain degree of respect and recognition.
And so, among those who find themselves frustrated with attending normal classes, if they've honed their skills in some field to such a degree that they dare hope a scout might notice—many have their fingers crossed, on the day that Hope's Peak sends invitations out.
(Temporary disclaimer: please avoid spoiling anything about Danganronpa, including the basic premise, to Celene or Truthshaper, until it comes up.)