How the percept works
When one tone is presented to one ear and a slightly different tone to the other, listeners can perceive a rhythmic beat corresponding to the frequency difference. For example, tones at 400 Hz and 410 Hz can create the perception of a 10 Hz binaural beat.
Why stereo separation matters
Because each ear needs a different input, headphones or a carefully separated stereo setup are normally used for genuine binaural-beat stimulation. If both tones are mixed together before reaching the ears, you are dealing with a different acoustic situation.
What binaural beats do not automatically prove
Hearing the beat does not by itself prove that the brain's ongoing electrical activity has been entrained, and entrainment would not automatically establish improvements in intelligence, memory or focus. Those are additional empirical questions.