Pillow lava is formed underwater, on the sea-bed. When the lava comes out and hits the sea water, the outside cools and freezes pretty much immediately, while the inside keeps on flowing. That means that it forms tube, which lengthens and widens until the pressure at the inlet end breaks open the tube and starts a new one. So you get the tubes interlocking.
This is how La Palma grew up from the sea bed. In most places pillow lava is under water, or under lots and lots of rock. But you can easily find it in the Caldera, where the river Taburiente has cut through it. That means you get a cross-section of interlocking tubes, which looks rather like pillows.
If you want to see this, take the road from Los Llanos into the Caldera in the direction of Brecitos, and park just before the ford across the river. Wander a little way upstream. [Obviously don't do this when the river is in spate, or when it's raining, or when it's raining on the peaks.]