Take a look at Paragraph 2.
I'll outline it in more or less plain english (I hope).
for 37 lakes they used this method:
1) they took a sample of water from 0.1 to 0.25 m below the surface. Using a thermos bottle. This would minimize change of the gas composition of the water with a change of temperature. The bottle was not sealed, but left out for a while in the air there at the lake (it was "equilibrated with ambient air"). If you were looking for issues you might or might not like this.
2) a sample of air 1m above the lake was taken.
3) they got a gas chromatograph to measure the CO2 in the air above the water sample, as well as in the air from above the lake. They might have taken the readings right there at the sampling point, or not its not clear. They may have also sealed bottles up and taken them to the lab. It sort of sounds like a point of sample measurement.
If the air above the water samples had more CO2 than the air above the lake, they called the water supersaturated as the water was emitting CO2.
This method does its best to simplify the question of whether fresh water bodies are saturated with CO2 and emitting it or if they might be absorbing it. By not outgassing the CO2 in the water and trying to get an absolute content of CO2, they are able to get around the various organisms and minerals in the water and pH which would vary the amount of CO2 the lakes can hold. they are just focusing on whether the water is giving off CO2.