As his starting point, Camus takes up the question of whether, on the one hand, we are free agents with souls and values, or if, on the other hand, we are just matter that moves about with mindless regularity. Reconciling these two equally undeniable perspectives is one of the great projects of religion and philosophy.
One of the most obvious—and on reflection, one of the most puzzling—facts about human existence is that we have values. Having values is more than simply having desires: if I desire something, I quite simply want it and will try to get it. My values go beyond my desires in that by valuing something, I do not simply desire it, but I also somehow judge that that something ought to be desired. In saying that something ought to be desired, I am assuming that the world ought to be a certain way. Further, I only feel the world ought to be a certain way if it is not entirely that way already: if there was no such thing as murder it would not make sense for me to say that people should not commit murder. Thus, having values implies that we feel the world ought to be different from the way it is.
Our capacity to see the world both as it is and as it ought to be allows us to look at ourselves in two very different lights. Most frequently, we see others and ourselves as willing, free agents, people who can deliberate and make choices, who can decide what's best and pursue certain ends. Because we have values it only makes sense that we should also see ourselves as capable of embodying those values. There would be no point in valuing certain qualities if we were incapable of acting to realize those qualities.
While we generally take this outlook, there is also the outlook of the scientist, of trying to see the world quite simply as it is. Scientifically speaking, this is a world divested of values, made up simply of matter and energy, where mindless particles interact in predetermined ways. There is no reason to think that humans are any exception to the laws of science. Just as we observe the behavior of ants milling about, mindlessly following some sort of mechanical routine, we can imagine alien scientists might also observe us milling about, and conclude that our behavior is equally predictable and routine-oriented.
The feeling of absurdity is effectively the feeling we get when we come to see ourselves in the second of these two alternative perspectives. This is a strictly objective worldview that looks at things quite simply as they are. Values are irrelevant to this worldview, and without values there seems to be no meaning and no purpose to anything we do. Without values, life has no meaning and there is nothing to motivate us to do one thing rather than another.
Though we may never have tried to rationalize this feeling philosophically, the feeling of absurdity is one that we have all experienced at some point in our life. In moments of depression or uncertainty, we might shrug and ask, "what's the point of doing anything?" This question is essentially a recognition of absurdity, a recognition that, from at least one perspective, there is no point in doing anything.