I am a Christian, if not a very good one, although I study other religions in order to understand religion's influence in countries where non-Christian faiths predominate, like in Afghanistan.
The burning of the Koran(s) at an American military base and the subsequently poor reaction of the Afghan population are an unfortunate reminder of cross-cultural differences that can be very dangerous if not understood, in order to be avoided.
The bottom line is, because of Islam's character as a religion as discussed below, if you want see a hostile reaction from a Muslim population, burning Korans is about the most predictable means of eliciting that response.
People die over such things, Afghans now and American soldiers, which isn't helping either party in the case of the current turmoil in Afghanistan.
As to why Muslims react so poorly to the burning of Korans, which makes Americans wonder about Muslim rationality in a way that is counter-productive so long as Muslims and Christians live on the same planet with nuclear and other very destructive weapons down the level of rifles and explosive suicide-belts, the reason is due to the overwhelming importance Muslims attach to the Koran as the revealed word of Allah-God via Gabriel to the Prophet Muhammad in comparison to the other monotheisitic faiths of Judaism and Christianity.
Its not that the Old and New Testament respectively are unsacred for the Jews and Christians of course, and the opposite, but in a comparative sense, there is a difference of emphasis between the three monotheistic faiths that if not understood, can get people killed, and potentially in large numbers.
It is a dicey situation as it is for the American Armed Forces and NATO to try to bring some sort of peace to Afghanistan, and such episodes, although clearly not intended as a deliberate affront, can also clearly have life and death adverse consequences to be avoided whenever possible.
As to the reason why burning a Koran elicits such a strong response, as for Christians and Jews of course, but even more for a Muslim as explained below, all that is holy in the Universe is contained in the literal words of Allah transmitted by the Angel Gabriel to Muhammad, starting with the basic command:
More than any of the other monotheistic traditions, especially in the Sunni branch of Islam, the largest branch of Islam in terms of adherents both in Afghanistan and in the Muslim world in general, the words of God-Allah are all that is needed for life to be lived.
Islam is therefore a faith that lends itself to a very austere approach to religion, compared to what we know in the West on average, even given the large numbers of commentaries on the Koran as to the Sharia law derived from the Koran.
In fact, much of early Muslim theological discourse was on the exact nature of how it was that Muhammad a mortal could interact with the Infinite in terms of God-Allah per the words themselves. That took centuries to sort out, and in the process cemented the comparatively extreme centrality of the recitation of the words of God-Allah that Muslims rightly or wrongly believe to have been given to Muhammed.
Given this basic religious orientation, and Koranic passages indicating a very austere interpretation of the Hebrew and then Christian traditions on idolotry, words in the Islamic tradition have more importance in a comparative sense than they do for the other main monotheistic faiths.
If you want an example that illuminates the centrality of words in Islam, Islamic art is a hint of this as to its very small amount of tolerance for representing even animals, as to do so would be idolotrous from their point of view, which also can be seen in the very intensive use of calligraphy in Islamic art, rather than representations of living things.
To burn a Koran then is in a comparative sense a grave afront to Muslims as to their faith, hence a predictably poor response, even though clearly in the recent case of the burning of the Korans in Afghanistan, it was tragically, given the uproar and deaths now, not intended to be an affront, but a "mere administrative convenience."
Frankly, given that this isn't really news to anyone who knows much about Islam at all, rather important in Afghanistan, the burnings were an error of the highest order under the circumstances so long as our mission is to bring peace to the long suffering people of Afghanistan.
But if you want to see why it happened as to the violent response in Afghanistan, an austere place as to faith even by Sunni standards as to actual practice, if you think about the deletion in Islam of things like iconography as in Orthodoxy, or the lack of stained glass as in cathedrals in the West, on average this reveals the comparative centrality of the words of the Koran in the Islamic faith, therefore it wouldn't be a surprise at all, if its of course completely horrible that this uproar has caused the deaths of American servicemen too, and endangered the good that all the sacrifces of the men and women of the American Armed Forces have made in Afghanistan to bring some sort of end to the Afghan people's long-running civil war.