To illustrate, Eagleman recounts the story of an illuminating experiment. To give you context, one area that has interested brain researchers for some time has been the fact that while our brain processes things as different speeds (sound faster than vision, for example), we are not consciously aware of this. Thus, when we see a batter hit the ball (assuming we are at close enough range), while we perceive the sight and sound simultaneously, the sound is actually available several milliseconds ahead of the visual. In essence, our brain "holds" the sound so it can present the two events together, thus constructing what it views as the most consistent narrative of reality.
So far, so good. Well, what Eagleman did was to set up an ingenious experiment to trick the brain, and in so doing expose this little ruse for what it is--simply another illusion presented as "reality." In the experiment, when subjects pressed a button a flash of light immediately appeared. At some point, the experimenters introduced a small lag effect, so the dot appeared a tenth of a second after pushing the button. However, after a few times the brain "learned" the delay, and the events once again appeared simultaneous. Then, the experimenters once again made them simultaneous, which caused the subjects to perceive the flash of light before pressing the button.
At first, this may seem a small issue--after all, who cares if our brains delay things by a tenth of a second here or there to make things more comprehensible? But this is only the tip of the iceberg (again, read the book...). Eagleman also talks of people who go blind but still fervently insist they can see, those who have mixed up senses (eg, they "experience" colors as tastes ), and an individual who developed a sudden and inexplicable interest in child porn that turned out to be due to a brain tumor (when they removed the tumor his interest went away; when he became interested again a couple of years later, it turned out they had missed some of the tumor).
The bottom line is that, for as solid as the "I" feels, and as much as we want to believe we are experiencing what is "truly" out there, such beliefs are nothing more than convenient fictions; lies, in fact, made vastly more believable since we tell them to ourselves.