The current best theory for the early universe at one point has matter coalescing from radiation. Such matter included mostly hydrogen, some helium, and a tiny amount of lithium. If you were to look at the periodic table of elements, you'll notice that these are the three lightest elements in existence.
These floating clouds of primordial matter collapsed under mutual gravitational attraction to form 'clumps'. On a small scale, these clumps became stars, and the collections of stars became first galaxies.
As the proto-stars collapsed enough to start fusion, the first step towards enriching the universe in heavier elements begun. Stars fuse hydrogen into helium, heavier stars take it further to the neighbourhood of oxygen, silicon and carbon, and the heaviest stars manage to fuse elements as heavy as iron and nickel. Fusion of heavier elements cannot produce energy, so it only happens when the heaviest stars explode in a supernova - the process releases excess energy that is partially consumed in fusing all the remaining (natural) elements of the periodic table.
The death of a star, be it as a supernova, or a less-violent expulsion of outer layers (like what our Sun will do), is the step where the heavier elements formed by the star are released back into space, where they get mixed with the primordial hydrogen and helium.
Over time, more and more heavier elements are produced, enriching the interstellar medium.
Since you, me, and the rest of the Earth are made from stuff that is not only hydrogen (helium never 'makes' anything - it's a noble gas, so it doesn't react with other elements), it means that most of the atoms we're made of were once made in the furnace of one or more dying stars.