Here are two verses from the Koran. Notice the portions which I have placed in capitals.
[Koran 2.191] And kill them wherever you find them, and drive them out from whence they drove you out, and PERSECUTION IS SEVERER THAN SLAUGHTER, and do not fight with them at the Sacred Mosque until they fight with you in it, but if they do fight you, then slay them; such is the recompense of the unbelievers.
[2.217] They ask you concerning the sacred month about fighting in it. Say: Fighting in it is a grave matter, and hindering (men) from Allah's way and denying Him, and (hindering men from) the Sacred Mosque and turning its people out of it, are still graver with Allah, and PERSECUTION IS GRAVER THAN SLAUGHTER; and they will not cease fighting with you until they turn you back from your religion, if they can; and whoever of you turns back from his religion, then he dies while an unbeliever-- these it is whose works shall go for nothing in this world and the hereafter, and they are the inmates of the fire; therein they shall abide.
Muslims issue death threats against people who insult their religion because they believe that the entire Koran, including these verses, is inerrant. They believe that persecution is really graver than slaughter. Therefore, anyone who persecutes Muslims by insulting their religion deserves to be slaughtered and anything less is showing them mercy which they don't truly deserve.
I have seen these verses quoted here even by Western Muslims who denounce terrorism (Proud Muslimah, for example), so it is clearly an important aspect of their religion. She used it as an example of the peaceful nature of Islam, but I really don't understand how anyone who believes that "persecution is graver than slaughter" can be considered peaceful.
Since I pointed that out to her, I have been blocked by most Muslims here from answering their questions ever since. It is amazing the extremes Muslims will go to to remove all dissenting opinions about Islam from sight. If their religion was really true, you would think that they would be capable of discussing it instead of blocking or killing anyone who disagrees with them.