Thanks for your question. "Study area" is defined as the region to which you can make inference from your study. In almost all cases, the boundaries of the study area are defined at the time the study is designed; in effect, defining the study area is the first
decision made when conducting a study.
For a survey to be representative of the study area, the boundaries of that area must be defined. After defining the study area boundaries, a randomised sampling scheme can be defined. The representativeness of the sampling scheme can be assessed by computer
means, see the
very first example
on our case study page.
The manner in which the size of the study area for the winter wren survey was determined can be seen from this satellite photo of the study area and adjacent agricultural land, shown in the example you cite:
The boundary of the wooded habitat can be seen in the image; the area within the open woodland can be calculated in a number of ways, perhaps using GIS software.
My main message is that the study area (hence its size) is determined prior to the establishment of a survey design or gathering of data.