"Burning" hydrogen is a phrase we use to describe what is happening on the Sun, but it is not like "burning" things on Earth -- "burning hydrogen" is a fancy way of saying the Sun FUSES hydrogen atoms into helium atoms. Nuclear fusion takes the particles that make up hydrogen and sticks them together to make helium (1 helium atom is made from 4 hydrogen atoms). In order to make the protons and neutrons in the helium stick together, the atom gives off tremendous energy, which makes the center of the Sun very hot and produces the light that we see here on Earth.
You can think of it this way, the Sun is very dense in its core and the atoms there are packed very close together. In addition, it is very hot and the high temperature causes the atoms to move around very quickly. The combination of high density and fast motion forces the hydrogen atoms to slam into one another quite often. When the hydrogen atoms slam together they release energy and produce helium. Once all the hydrogen is used up the Sun will start to fuse helium in the same manner, producing carbon as a by product.
You could imagine packing a bunch of people into a small room and then having them run around very quickly. They would not be able to do that without bumping into each other. The faster they move, the harder they will slam into one another. Each slamming together releases energy and within the Sun that energy is what fuels it.
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