Spacers essentially do exactly as the name implies, they add 'space' between the face of the wheel, and the hub it normally is bolted up to. This gap however small (usually in increments of 5mm, 10mm, 15mm, 20mm, 25mm, etc.) can improve both the looks and the handling of your Z very much.
Aesthetically, it will bring your wheels/tires flush with the body of the car, hence eliminating that horrid gap and overhang the 370 suffers from. Some choose to go as far out as legally possible and make them flush. Others choose to just have it be farther out, still being covered by the fenders. The style is purely up to the driver.
Performance wise, moving the wheels out in this manner increases your wheelbase and helps lower your center of gravity in your car. This translates to having an easier time maneuvering around corners. A wider stance means your car is less likely to roll as you move the wheel from side to side. To help compliment this, its also common for people running spacers to get lowering springs and sway bars. This combination of lower/wider/stiffer will drastically improve your handling for a much more spirited drive.
If you thought your Z was awesome before, just wait till you can take the same corners you used to at 65 and feel squeemish, when your taking them at 85 and having this huge grin cause your car won't budge.
All you need now is a good set of tires and you'll be ready to tear up the streets after graduation.