A membrane is a lining that allows some things to pass through but not others. In a hatching egg a membrane develops around the developing chick that protects it from touching the inside of the porous egg shell but it allows air to pass through. The chick needs fresh oxygen and needs to get rid of CO2. The membrane allows those to pass through. It also allows moisture to escape from the egg. If the developing chick touches the inside of the porous egg shell it can get stuck, which would keep it from hatching. The membrane stops that. But if the membrane dries out too much it can shrink tight around the chick and prevent it from hatching. We call that shrink-wrap.
As incubation progresses the egg loses some moisture and the air cell in the fat end grows. You should be able to see that if you candle the egg. When the chick is ready to hatch it pokes a hole in the membrane separating the air cell from the rest of the egg and uses that air to learn to breathe in an atmosphere instead of living in a liquid environment. Poking the hole in that membrane to get into the air cell is called internal pipping.
