Because the decimal expansion of 19 repeats after 18 digits.
That means all the fractions 1/19 .. 18/19 are just rotations of each other. And 20/19 = 1 + 1/19, which means that 2/19 is a single rotation away.
If you've ever noticed that 142857 * [1..6] produces different rotations of the same number and wondered why, you might have thought about this. Dyson had probably already done all the hard work on the problem and knew immediately that the answers were rotations of some repeating fraction, and one where multiplying by 2 gives rotations by 1 digit.