I own one of those systems. The green device gets connected to a faucet that is properly threaded, sometime you just have to take off the aerator on the faucet. The green device is simply a venturi, it has the point that connect to the faucet, to the hose, and an outlet on the bottom, the bell like structure at the bottom.
When the water is turned on at the faucet is runs past the point where the hose connects, this pulls the water from your aquarium into the sink. If you close the bell like structure on the end of the green venturi then the water will flow from the faucet into the aquarium.
With the system you do not need any buckets because the water goes directly from the tank to the sink or from the sink directly to the aquarium.
Using cold water only I would run the system in syphon mode to pull the water out of the tank. Once you have removed enough water from the tank, close the valve on the tank end of the unit and shut off the faucet water.
Get enough water treatment ready to put into your fish tank based on the volume of water in the full tank, go to the Venturi end and adjust the water temperature to what the water temperature is in the tank. Then double check that the valve close to the gravel cleaner is close, then close the bell like structure on the venturi so that the water goes towards the tank. Add the water treatment to the tank then open the valve on the tank side to let the water flow in.
The system is good but wastes some water, not good if you are on a septic field. Also be cautious, the python hose seems strong enough for the vacuum when syphoning but I don't know how much pressure it can hold when filling.