I was wondering, what exactly does the beam angle of a floodlight measure?
The beam angle tells you what area in front of the flood light will be illuminated.
The brightness of the flood light is measured in lumens. This is taken at the centre of the flood light's beam where it is brightest, and then you keep on angling out until the brightness is 50% of the centre. You double that angle (because the flood light obviously shines to the left and right of the centre) and that is the beam angle.
Correct - that is the standard way to calculate it, but it worth noting that the light continues to spread out past the beam angle, into what is known as the beam field, which is the angle until the brightness is 10% of centre brightness.