Stepper motors require some voltage but they really depend on current, you can supply them with 3, 6, 12, 24.36,48 as long as you don't exceed the current capability of the stepper. You can use most steppers that work at 12v on 24v as long as you set the current limits correctly.
If your board can accept 24V then usually it also using 24V to drive the steppers, it doesn't know that the steppers are any specific voltage , only what current they need based on the supply voltage.
The stepper controller slices up the power to the stepper to keep the current within the limit for the stepper. Why a higher voltage matters is that during this slicing up of power , to energize the stepper the first pulse or series of pulses is usually the full power supply voltage giving the stepper greater starting torque . Higher voltage lets you accelerate faster and also brake faster around corners since you have more holding force in the motor. Once the motor has started to move then the stepper controller will continue to lower the voltage in each pulse to keep the current level below the limit.

The cost of higher voltage is heat on the motor and controller. Most commercial machines use 48V and higher .