How were underwater volcanoes able to erupt frequently enough to create islands, instead of the water cooling down the magma and clogging them?
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The water *does* cool down the magma. That’s what becomes the island: the cooled down magma. But it won’t cool down the source (deep in the earth) and it won’t “clog” it because it’s moving. Or rather, the crust of the Earth moves over the hot spot, which is why these islands end up as strings of islands.
If you’re asking why the water doesn’t always cool the lava down and always prevent eruption, there’s a complicated answer, but I’ll try for an appropriate ELI5.
Water can only cool lava down so much. The two materials, lava and water, will try to “average” their different temperatures. The lava will heat the water exactly as the water will cool the lava.
Using a completely made up number of 4000 degrees for lava and 50 for water, one cube of lava will cool down to around 2000 degrees, but a cube the same size of water will heat up to the same, which will boil it instantly. That’s only a small part of the lava, and a small part of the water.
For the lava, it will be heated again by the lava behind it, some will stay solid, but some will heat back up. And for the water, it will escape as steam, which has a side effect of creating a low pressure area where there isn’t any water (or as much).the lava under pressure with rush to fill that gap made by the steam.
This cycle is repeated on a larger scale covering huge volumes of water, with huge amounts of pressure, meaning that maybe for a while, the lava is refrozen and turned into rock, eventually the lava overcomes the waters ability to cool it, and overcomes the pressure keeping it down, and we get an eruption.
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