I just paid $16 per tire to have them balanced. I know I have to do this, but I don't know why. Why are those counterweights on there?

You were overcharged, first, most new tires at MOST shops, balance for free.

Tires are manufactured and are "close to" balanced, but not perfect. On the new tire, there's a spot of paint. This "dot" is the spot where a small amount of rubber is omitted in order to make up for the stem and the weight of things like the tire sensors. IF the tire shop lines this paint dot up to the valve stem, they will use a much smaller weight. Most shops won't take that extra time and effort in order to get the balance close.

Here's a good recommendation. When they are being balanced, insist that the weights are mounted on the INSIDE of the rim (wheel) rather than on the outside where you see it. First, with full wheel covers, the hub caps sit better and are tighter. (You see cars with hub caps missing all the time, this could have helped to prevent that) Secondly, on custom or fancy wheels, there is a clear-coat on the wheels and the weight being hammered on will damage that coating. And the weight showing takes away from the look of the wheels.

In addition to providing a smooth ride, balancing is a key component in tire wear. If the wheel is not balanced the tire would not adhere to the road surface properly. It can literally "bounce" off the road causing but uneven and premature tire wear.

the wheel weights are on a rim because tires from the factory are never 100 % round; therefore, the weights are installed to help with the balance of the wheel/tire ............w/o weights, the tire will bounce when driving

Just as the name suggests, so they are balanced. Otherwise your tires may shake going down the road and your tires will wear quicker.

Out of balance tires cause increased wear on steering, suspension, wheel bearings, shocks, and tires.

One out of balance tire can ruin a shock absorber in a few thousand miles.

to balance the tire and rims.