The Ben Ish Chai (a”h) states that one should open the Megilah scroll in its entirety before the reading, specifically, before the berachot are recited over the reading. However, Ha’Rav Ovadiyah Yosef explains the Ben Ish Chai’s comments as referring specifically to the Chazan, who reads the Megilah for the congregation. He should open the entire scroll before the reading, and keep it open with one part folded over the other.
The congregants, who listen silently to the reading while following along in their own scrolls, do not unroll their Megilot before the reading. They keep it closed and open it as the reading progresses, leaving it open until the conclusion of the reading.
After the reading, they roll the Megilah.
The reason for this minhag is the fact that Megilat Esther is referred to as an “Igeret,” (a “letter” or “document”), as opposed to a Torah scroll, which is never described in this term. Letters are usually left wide open as they are read, and it is therefore customary to keep the Megilah open (whereas a Torah scroll is kept closed except for the page currently being read).
However, as mentioned, the Chazan opens the Megilah in its entirety before the reading, whereas the congregants open the scroll as the reading progresses.