A transistor to the output pin of the Arduino to switch a larger load (likely the 9-12V source powering the Arduino) that in turn switches a relay that can carry the current of the brewer...
It's main level voltage and current that can be deadly and can also cause a fire or other dangerous situations, caution and safety should be forefront... Since you state it's your first experiment may I suggest you take a step back and start with several smaller less lethal and dangerous things...