You don't need most anything but water and lights. A skimmer is a tool that will give you a bit more latitude with regard to tank maintenance and water quality. Many people run fantastic SPS tanks with no skimmer. A skimmer basically removes DOC or disolved organic carbon, in other words, wastes. It removes wastes before they get to the bacteria to be converted to less toxic chemicals like nitrate. So a skimmer will (if it is running correctly) reduce the workload on bacteria which should reduce nitrate accumulation.
A skimmer does not remove trace elements except in the small amount of water it removes, but it will not target these elements.
It works because most (but not all) DOC is made of a certain molecule which has two parts, one end is attracted to water and one end is attracted to air. Due to this, these molecules love to be on the surface of a bubble where it can be in both water and air at the same time. The surface film of a bubble is the perfect place to attract DOC. The bubbles burst at the top of the reaction chamber because when they go over the rim of the reaction tube, there is less preassure on the sides and they burst, dumping their contents into the cup.
I myself use a large skimmer and have for decades but there are people who feel that a skimmer actually hinders SPS by removing some things that SPS "likes"
