The following is clearer than anything I have ever heard.
The term anatman is usually translated as "non-soul", but in reality atman is here synonymous with a personality, an ego, a self, an individual, a living being, a conscious agent, etc.
The underlying idea is that, whatsoever be designated by all these names, it is not a real and ultimate fact, it is a mere name for a multitude of interconnected facts, which Buddhist philosophy is attempting to analyse by reducing them to real elements (dharma). Thus "soullessness" (nairatmya) is but the negative expression, indeed a synonym, for the existence of ultimate realities (dharmata).
Buddhism never denied the existence of a personality, or a soul, in the empirical sense, it only maintained that it was no ultimate reality (not a dharma). The Buddhist term for an individual, a term which is intended to suggest the difference between the Buddhist view and other theories, is santana, i.e. a "stream ", viz. of interconnected facts.
It includes the mental elements and the physical ones as well, the elements of one's own body and the external objects, as far as they constitute the experience of a given personality. The representatives of eighteen classes (dhatu) of elements combine together to produce this interconnected stream.
There is a special force, called prapti, which holds these elements combined. It operates only within the limits of a single stream and not beyond. This stream of elements kept together, and not limited to present life, but having its roots in past existences and its continuation in future ones - is the Buddhist counterpart of the Soul or the Self of other systems.
From Theodore Stcherbatsky in 'The Central Conception of Buddhism and the Meaning of the Word "Dharma".'