Actually it is related to logs and exponents.
It is typical to consider 1/2 value, e.g., half life, half-power or half of some signal that decays exponentially.
For an exponentially decaying signal, one interesting property is that half-signal value is constant; it does not depend on the original reference point.
If you have an exponentially decaying signal, and you plot that on a half-log paper (x-lin and y-log), you get a straight line.
If you also convert the y-axis scale, 1/2 point will be log(1/2)=-log(2)=-0.3010 or -3db. That is where the 3db comes in the picture.
Hence, -3db point is the one where the signal becomes 1/2; at -6db point the signal becomes 1/4; at -12db point the signal becomes /16th of the original.
Next question will be the order of the filters. Well, that is another story.