The Shulchan Aruch in Siman 690:17 brings down a minhag klal yisrael to fold the megillah like an iggeres (Esther 9:26,29) when reading from it. The Mishnah Berurah mentions that one should not pause excessively by taking a breath in the middle of a verse, so as to read like an iggeres. I've seen iggeres translated as "letter", but these attributes don't seem to apply to a letter as I understand it. Perhaps letters were written completely differently back then, but in that case the translation "letter" is inaccurate. So what is an iggeres?
My understanding (which I'm half certain I saw in a sefer many moons ago) is that just as when one receives an important letter, one reads it avidly all the way through without pausing, so too should the Megillah be read. Therefore, it should be read without pausing unnecessarily.
The reason for unfolding it before the reading is because the Megillah has to be in the form of a sefer, which in the language of the Torah means a scroll, and so all the pages are joined together to make a sefer. But the reading should be done like reading a multi-page letter (for the reason mentioned above), and since we can't separate the individual pages, we unfurl it and fold it page over page so that it becomes like a multi-page letter and not a sefer.
An alternative explanation as to why we fold it is because the main thing is to unfurl the scroll completely so that appears to be a letter and not a sefer. But since the Megillah is very long this will usually result in it either touching the floor or at the very least hanging off the edge of the table, either of which is a disgrace to the sefer, as the MB writes in Se'if Katan 56, and therefore we fold it.
It appears the Megillah is referred to as both a "sefer" and an "iggeret". The Gemara in Masekhet Megillah 19A (Hebrew, English) discusses this matter as referenced in this shi'ur from Yeshivhat Har Etzion.
It appears from this shi'ur and the Gemara that the word "iggeret" does, in fact, translate into English as "letter"; but, that the particular sefer/iggeret of Megillat Esther should be bound by three sinews of a kosher animal. This would help to explain the Shulhhan Arukh referenced in the question.