There is a Zen saying that states “before enlightenment a mountain is just a mountain. As one begins on the path toward enlightenment, a mountain is no long a mountain. When one becomes enlightened, a mountain is just a mountain again.”
Anatma is what the concept of "no-self" is called in Buddhist psychology. In earliest forms of Buddhism, the notion of self was viewed as an illusion and thought of as an obstacle to Truth (later forms of Buddhism incorporate the idea of reincarnation which conveniently seems to "forget" that the idea is nothing more than a subtle form at clinging - to the "idea" of self, but I digress).
I once read somewhere “there is a razors edge between enlightenment and despair” (I forget who said it). The concept has always stuck with me though, because it so clearly described my frustration at trying to explain the concept to others, especially when people ask me (which they CONSTANTLY do) how I can be "okay" with not believing in a soul or heaven, etc. For me, it's always been a hard thing to convey to people, as the many moments of deep clarity often experienced during formal meditation tend to spring from “experiential knowing,” which speaks to the holistic right hemisphere - the side of the brain that experiences the world but doesn't have a voice (speaking and logic - as well as our propensity to divide things into categories and apply distinct labels to things - come obviously from the left side of the brain).
Anyway, the phrase spoke to me because I believe it does a good job of explaining how one can easily become overcome with despair at the idea that there may be nothing more to ourselves then "dynamic, ever-changing manifestations of potential" - hence no soul).
For me though, the other side is the realization can exhilarating. Many don’t understand how this is possible, and I do a horrible job of trying to explain it. I believe however, that it can be remotely summed up in the idea that "when a man/woman realize that he/she has been changing continuously every moment, he/she grieves neither for what he has lost nor for what he has not gained." It's a hard concept to grasp for many (especially without a formal meditation practice). Most people tend to get overwhelmed or depressed by such a concept, especially Westerners, and especially if the concept of a soul is holding together an ego fractured by intense trauma compounded by grief and loss of losing people who are close to you (which just happens to be many of the people closest to me).
One of my favorite Zen quotes is "it takes a long time to understand nothing." This speaks on the concept of emptiness of all things (because everything in the world is constantly changing, there is nothing to hold on to - everything that we perceive with our senses that seems so solid and enduring merely maintains a “façade” of permanence - everything, with time, is subject to decay and transformation). When working with people who have experienced trauma in their lives, therapists often speak of complexity theory, and the fact that trauma causes ones linear progression of becoming (increasingly complex) to hit the pause button, so to speak. The concept of emptiness may seem threatening to such a person. Hence, people can practice Buddhism without having to buy into the concept of Anatma but, like anything else in life, it comes with a price (and one might say, with a “pause button”).
You know, it's interesting. I really think that us Americans are still stuck with our inherited Platonic idealism. Plato's concept of "essentialism" states that for any specific kind of entity, there is a set of characteristics or properties all of which any entity of that kind must possess - a generalization stating that certain properties possessed by a group (e.g. people, things, ideas) are universal (there is an quantifiable “essence” in everything - of what it means to be a "rabbit" or a "human", etc). From a evolutionary standpoint, this concept is ridiculous, as there is no essential form of "rabbit" or of "horse" and certainly not "human. Hence, as Dawkins often states, if you were able to take every fossil from your ancestry and lay it out in time all the way back to the earliest discernible mammal and follow the line all the way back to the present - through primates and up to your parents and finally to you - you would have an UNBROKEN LINE of descendants and the thus idea of essentialism would inevitably break down because it would be impossible to draw any distinct lines between differing species.