How did the earth's atmospheric blanket form?

Well, the atmosphere was initially comprised of gases that were left over after the Earth formed that were kept around the Earth due to gravity. Originally, the atmosphere had little to no oxygen. The atmosphere we have today is very oxygen-rich because when plant life evolved, they evolved in such a way as to take advantage of the large amounts of carbon dioxide in ancient Earth's atmosphere, which is why photosynthesis takes in carbon dioxide. And since the waste product of photosynthesis is oxygen, over millions of years all the plants on earth produced enough oxygen to greatly increase the atmosphere's oxygen content, allowing for the evolution of oxygen-dependent life forms like humans.