Crater Lake was formed when Mount Mazama erupted thousands of years ago. As the massive magma pocket released its contents, throwing volcanic rock as far as Boise, Idaho and Canada, the action created an empty space above the magma pocket. No longer supported, the top of the mountain collapsed in on itself.
The collapse created a large bowl like depression, many thousands of feet deep. However, the magma pocket wasn’t empty, it was just partly drained, so the mountain was still volcanically active. There was still magma welling up from below. This worked its way up through the rocks of the collapsed mountain, creating cinder cones.
Cinder cones are common in the Cascade Mountains, and several were produced on and near Mount Mazama. One of them was what was to become Wizard Island as the water level in Crater Lake grew. It is off center in the lake, and on the top, a depression can be clearly be seen from the rim of the lake, that is quite similar in appearance and cause as that of the caldera that contains the lake, except that it is much smaller and shallower.
Much of Wizard Island is made up of pumice, virtually the froth off of magma that has turned solid. Cinder rocks are denser, but also can be found on the island. Various plants and trees now grow there, but that most likely took place long after the main part of the formation was long over.
It is likely that Phantom Ship was created in the same way, though there is some disagreement about its formation.
Over thousands of years, Crater Lake filled. Originally, it was thought that this happened only because of precipitation, since there is no known inlet or outlet to Crater Lake. However, in the 80s and 90s, studies showed that there are also geothermal springs at the bottom of this very deep lake that may have helped to fill it with water.
Today, the lake level stays stable, with the evaporation being offset by the snow and springs in the lake. Rain is uncommon at the lake, but snowfall definitely isn’t.
However, in order for Wizard Island to be an island, the water was necessary. At some point back in history, it is likely that there were either other sources of water that were draining into the lake, or a greater amount of precipitation or inflow from the thermal vents. Even at this, it took a great deal of time to make Wizard Island an island.
The volume of water in the lake is staggering, yet how the island formed isn’t in question anymore. It is a dominating feature of the lake, and the deepest known part of Crater Lake isn’t far from the shore of Wizard Island. The thermal vents also show that there is still a magma pocket somewhere below the bottom of the lake.
This means that Wizard Island may not be finished forming or evolving. It could conceivable be joined by other islands sometime in the future. How it was created isn’t much of a mystery. What will occur in the future definitely is.