I've heard multiple explanations from this, ranging from accusations against Julius Caesar starting a great fire, to Omar Ibn Al-Khattab ordering the destruction of all secular and infidel books held within.
I've also heard those who say that describing the event as a "destruction" or a "burning" is inaccurate, rather that the library slowly fell into an unkept, unstaffed, derelict state due to underfunding, neglect, and a brief bout of scholastic-related hardship in Alexandria.
My college textbook Landmarks in Humanities 4e says the following, " Alexandria, which replaced Athens as a cultural center, boasted a population of more than one million people and a library of half a million books (the collection was destroyed by fire when Julius Caesar besieged the city in 47 b.c.e.)" (Fiero).
Thing is, not only did it burn down several times, whatever people may be meaning with the term may actually be one of several different buildings, some of which burned down at different times...so, which time? u/XenophonTheAthenian covers which structure, and which time (also along with u/EyeStache), and a bunch more relevant facts about the whole place. u/Naugrith also has a relevant post.