If a stethoscope is placed over the brachial artery in the cubital fossa in a normal person (without arterial disease), no sound should be audible. As the heart beats, these pulses are transmitted smoothly throughout the arteries and no sound is produced.
If you place the cuff of a sphygmomanometer around a patient's upper arm and inflate the pressure above the patient's systolic blood pressure, there will similarly be no sound audible. This is because the pressure in the cuff is high enough such that it completely occludes the blood flow. Think of a flexible tube or pipe with fluid in it that is being pinched shut.
Now, if the pressure is dropped to just slightly below the patient's systolic blood pressure, the first Korotkoff sound will be heard. As the pressure in the cuff is slightly below the pressure produced by the heart, some blood will be able to be passed through the upper arm when the pressure in the artery rises during systole. This blood flows in spurts as the pressure in the artery rises above the pressure in the cuff and then drops back down, resulting in turbulence that results in audible sound.
As the pressure in the cuff is allowed to fall further, thumping sounds continue to be heard as long as the pressure in the cuff is between the systolic and diastolic pressures, as the arterial pressure keeps on rising above and dropping back below the pressure in the cuff.
Eventually, as the pressure in the cuff drops further, the sounds change in quality, then become muted, then disappear altogether. As the pressure in the cuff drops below the diastolic blood pressure, the cuff no longer provides any restriction to blood flow allowing the blood flow to become smooth again with no turbulence and thus produce no further audible sound.
Korotkoff actually described 5 types of Korotkoff sounds. The first Korotkoff sound is the snapping sound first heard at the systolic pressure. The second Korotkoff sound are the murmurs heard for most of the area between the systolic and diastolic pressures. The third and fourth sounds, at pressures within 10mmHg above the diastolic blood pressure were described as "thumping" and "muting". The fifth Korotkoff sound is silence as the cuff pressure drops below the diastolic blood pressure.
The systolic blood pressure is taken to be the pressure at which the first Korotkoff sound is first heard and the diastolic blood pressure is the pressure at which the fourth Korotkoff sound is just barely audible.