With a blood pressure cuff, what you are doing is collapsing the arterial wall with outside pressure. You increase the pressure above systolic blood pressure and then gradually reduce it until you hear a sound. This sound isn't the heartbeat, but is instead the result of turbulent flow in the blood vessel. While you are between systolic and diastolic pressure, only spurts of blood can flow through the vessel creating noise. Eventually, you reduce the pressure enough and blood even during diastole (which is lower pressure) can expand the walls of the artery against the blood pressure cuff.
So, in both cases what you are hearing is due to blood pushing open the vascular walls.
So that's how doctors do it with a manual pump and stethoscope. Never thought of asking one ...
When you usually measure blood pressure you increase the pressure to a value that's likely above the systolic value of the patient (say 180). Then you gradually release pressure. Once you hear a heartbeat that's the systolic value. Once you stop hearing the heartbeat that's the diastolic value, which is the vascular walls resisting the heartbeat
In this particular case instead of the usual heartbeat you'd likely hear something more like a constant flow but you'd still have systolic and diastolic values.