How does the blood pressure cuff measure blood pressure?

See below. The cuff is put around the arm and inflated. The pressure is increased enough to totally cut off flow through the artery. When the cuff is deflated, sounds are heard when flow begins. That is the systolic pressure. When the cuff further deflates, eventually there is no collapse of the artery at all and then the sounds disappear or muffle. That is the diastolic pressure.
Blood pressure . Using air pressure in the cuff the artery in your arm is compressed and flow is halted . As the air is bled out if the cuff the artery "opens" and a sound us heard that represents the systolic pressure as the cuff is bled some more the sound eventually disappears giving the diastolic pressure.