You are looking at photometric methods, so not at a spectra. All you can measure is light intensity. By placing a filter you decide the wavelength of the light you are measuring the intensity of.
AS the black body curve is not flat but curved, different wavelengths from the same black body will will have different intensities. The difference between these is dependent on the temperature of the curve, due to the shift of the maximum wavelength. Thus by measuring the B-V index you can determine a stars class.
The reason they use B and V is because of the sensitivity of the traditional means of detecting light. B is the wavelength film picks up the best. V is the wavelength the eye picks up the best. They could have use other wavelengths and do.
I'll have to learn to type faster.