That article by Ven Than may clear things up, but I think it is important to say 2 things about anatta right from the start. These are the two issues that I've encountered when talking about anatta with people interested in Buddhism:
Anatta does not mean that "everything is an illusion" in an ontologically nihilistic way, that all this stuff is not actually occurring. "No-self" is sort of a misnomer that can lead to this idea, I think. "Not-self" being the preferred term. Since we tend to naturally identify with our experiences, when we hear that there is no self, we then think, "Oh, then everything is an illusion?" But that logic is based in a false assumption, that these things are "us".
The things you experience are really happening (a term I'm using to avoid a dichotomy of is/is not real/actually existent/true etc.), it is the conceptual overlay of self and the related concepts of I/me/mine that are said to be false, because in the flow of our experience there is no self-existent nugget of selfhood that can be found. Larger metaphysical statements of knowledge of selfhood-or-not are beyond the range of experience, but we can see in sight, sound, touch, taste, smell, and cognition, that there is not actually any "self" that we can pinpoint. So these things are not-self.
And on the other hand, anatta does not posit a kind of transcendental hivemind-esque conscious eternal monad that we are all a part of which is the real truth of ourselves (this relates to tattvamasi, I believe), that is taking things too far. It's just that there's no self that we can find. And yet continuing to habitually add a self into the mix causes us stress. Everything we experience is unreliable, impermanent, prone to bringing us dis-ease. When this is understood deeply, one stops craving for this or that, seeing the pointlessness of it. Being dis-en-passioned about things, one doesn't get tied up them, doesn't fight against them, no longer has conceptions of a "me" getting drug around in it all. Being like that, the mind isn't agitated or consumed by what is pleasant or unpleasant, and so is peaceful.
Hopefully what I've said makes sense and is not unhelpful blabber!