I apologize if you are anyone else already knows this, but I am trying to help,
The stocks style works like this, the wastegate has an internal spring inside a round metal housing that is pulling on the rod that you see extended out from the housing. Grab the rod with your hands, you can pull on the rod compressing the internal spring. Connected to the other end of the rod is lever arm that disappears into the exhaust outlet of the turbo. Connected to the lever arm is a flap "puck", this puck covers up a hole in the exhaust housing that is directly bolted to the exhaust outlet of the turbo. The function of the hole in the turbo is to allow exhaust energy to bypass the turbine of the turbo thus not provide any energy in spinning the turbine blades, thus not spinning the compressor side of the turbo, thus not creating boost. If this hole was not covered by the puck at all, the exhaust (energy) would have two paths to flow into the downpipe. One being through the turbine blades thus spinning the turbine blades in the process, the other is through the bypass hole (wastegate port) which would not spin the turbine blades. I guess that is why it is called a WASTEgate lol. Anyway, The stock wastegate is pulling on the lever arm keeping it closed. This forces all of the exhaust through the turbine blades which creates airflow on the cold air side of the turbo (compressor). This flow of air eventually fills the engine and starts becoming pressure (boost). Meanwhile the exhaust pipes are filling with air and pressure is building as well (back-pressure). This pressure starts to really push hard on the puck. When the pressure is great enough it overpowers the springs ability to keep it closed and thus opening the wastegate hole. A portion of the exhaust gas exits through this hole and does not add to the force that is driving the turbo. The net affect of this is the boost decreases in the intake manifold.
The purpose of the black vacuum hose going to the wastegate port is to actually help the exhaust push open the spring. It helps to compress the spring closed from the opposite side that the exhaust pressure is pulling. Pressure is coming from the turbo and is being directed into that round metal housing of the wastegate. This helps compress the pressure spring and helps the exhaust open the puck. Think of it as someone pulling from one side and as someone pushes from the other in a team effort to compress the spring. If you take the vacuum hose directly to the wastegate port, the result will be lowest possible boost setting the engine can have. This is derived from the amount of pressure it takes to compress the spring. The stock wastegate has spring that is more easy to compress than an aftermarket heavy duty one. Stronger the spring in the wastegate the more force the exhaust has to exert on the puck to open it before it can bypass the blades of the turbine, resulting in a lower intake manifold air pressure (boost). When the hose is disconnected from the wastegate port the result will be the highest possible engine boost. A lot of times an adjusting valve will be placed in-line and so you can adjust how much help the pressure of the intake manifold will help in lowering the boost (pushing on the spring, helping the exhaust open the puck).