It isn't completely constant over geologic time. Earth's oxygen has been as low as 0 and as high as about 33% at different points during Earth's history.

The oxygen in the air is the result of an equilibrium between plants and other photosynthetic organisms (it's actually mostly microbes in the ocean) producing oxygen as a by-product of photosynthesis. This equilibrium is pretty stable, since organisms are short-lived and organisms will be slightly more successful in environments that are more favorable to them. (As an example, global plant growth has accelerated pretty significantly as humans have added carbon to the atmosphere, although not by enough to compensate for the rate at which we're adding it.) This part is basically a closed cycle: plants produce oxygen in the process of capturing carbon to use in their "bodies", animals eat the plants and burn the plants' "bodies" with oxygen from the air.

Over the very long term, though, there's another important effect: geology. Or more specifically, geology driven by living things. Earth is old enough, and living things dominant enough in its surface chemistry, that life on Earth and Earth's own geology are intimately linked over the longest timescales. (This is, I think, incredibly cool. In a very real sense, the very stones of Earth are part of the biosphere.)

In the processes described above, some organisms manage to die and decompose without returning their carbon to the atmosphere. The corresponding oxygen that was liberated during the capture of that carbon, then, ends up sticking around in the atmosphere over the long term. This is, in effect, what happened during the burying of the organic material that became fossil fuels, and it's part of why the era in which those fuels were produced had the highest oxygen levels in the history of Earth to date.

On the other hand, sometimes geological processes expose this buried carbon to the atmosphere, where it reacts with oxygen. This is the same kind of chemical cycle as the regular biological one, just dominated by geological processes over much, much longer times.

There are a few other minor contributors, like serpentization of certain rocks, but the ones described above are the big drivers.

The net effect of all of this in the Earth we have today is very close to zero if you ignore humans. So, human activity aside, Earth's oxygen levels aren't changing quickly over any timescale remotely relevant to humanity.

But thanks to the fact that we're burning an absolute ton of buried carbon (effectively accelerating one side of the geological processes described above), the oxygen content of the atmosphere has very slightly declined due to human activity. Not by anything significant - a few parts in 10,000 - but some.