Space suits have a cylinder of fresh oxygen that the astronaut wears along with the suit, along with breathing tubes. This allows the astronaut using the space suit to breathe. The breathing setup also recycles the gas exhaled by the astronaut.

Suits worn by people swimming underwater also have such breathing arrangements.

The space suit has its own air supply within it that is kept within it so that it can't leak out. You breathe the air inside the suit.