An electronic compass - mounted somewhere on the airframe that is relatively free of interference from electrical & magnetic fields - sends a signal to electric motors in the HSI. The motors move the compass card/gyro combination of the HSI to keep it in sync. with the remote compass. This is the slaving to which you refer. It removes the need to periodically manually set the compass card of the directional indicator/HSI to the wet compass.
In case there's problem with the signals from the remote compass the slaving can be disabled so the HSI's compass functions now behave like a standard, manually set directional indicator. The electrical motors are now manually controlled with a couple of buttons to set the compass card to the wet compass.