Here's the chemical formula of fat.

The planet's gravity holds the soil, water, and air around the planet so nothing "drains out" into space. So everything gets recycled.

You've heard of plants photosynthesis? That's this reaction, carbon dioxide + water -> sugar + oxygen (released). The plant then chains the sugar together into cellulose which forms the body of the plant.

Animals eat plants and they digest the cellulose back to the individual sugars. Then they breathe in oxygen, and do the reaction in reverse: oxygen + sugar -> water + carbon dioxide + energy. The animal then pees the water and exhales the carbon dioxide.

So water (hydrogen and oxygen) and carbon dioxide (carbon and oxygen) get recycled around and around the Earth, for billions of years, between plants and animals and plants again and animals again.

Your body is made mostly of long chains of carbon + hydrogen and some oxygen here and there, like that fat linked above.

The term "carbon-based" is used as a contrast to "silicon-based". Carbon has this property of being able to link to itself in arbitrarily long chains, and thus form complex organic molecules. The (only) other element that can do this is silicon. Here's an article that discusses silicon-based life.