Question
I am reading that gardeners often speak of outdoor plants having become "established," what does this mean and at what point does a plant have to cross in order to become established and no longer require human care?

Answer
It simply means that the plant has been growing in its new location long enough to have grown new roots that meet its water and nutrient needs and it no longer needs special attention.

The way you can tell that this is occurred is that you can no longer pull the plant up, not that I recommend trying since, if the plant is NOT yet establish, you will damage the developing root system.

As to how long the process takes, that depends on the variety of plant and the interim care it is receiving.  It can be weeks or months, up to a full growing season.