Boyle's Law states that: the Pressure of a fixed mass of gas is inversely proportional to its volume, if its temperature is kept constant.
In the process of breathing-in, our chest grows slightly and the rib
cage kindoff gains in size(and volume), and so the air inside our lungs
occupy a larger space(volume).
This increase in volume causes the Pressure of the air inside our
lungs to become smaller than the pressure outside( in the atmosphere
around). So, air is pushed in due to the pressure difference that results.
When you breathe out, the reverse happens. Your rib cage(and lungs)
contract and so the air in them has a smaller volume. And so, the
pressure increases.
This time the pressure difference is towards the outside. And so, air is pushed out!
