Why 1 and 5? Because they feel right.
I'm sure there are studies on the emotional value and cognitive salience of specific numbers, but we can understand the choice of 1 and 5 without having to resort to research.
The people that created today's statistics were born, raised and live in a decimal world. Of course there are non-decimal counting systems, and counting to twelve using the phalanges is possible and has been done, but it is not obvious in the same way as using the fingers is (which are therefore called "digits", like the numbers). And while you (and Fisher) may know about non-decimal counting systems, the decimal system is and has been the predominant counting system your (and Fisher's world) in the past hundred years.
But why are the numbers five and one special? Because both are the most naturally salient divisions of the basic ten: one finger, one hand (or: a half).
You don't even have to go so far as to conceptualize fractions to get from ten to one and five. The one is simply there, just as your finger is simply there. And halving something is an operation much simpler than dividing it into any other proportion. Cutting anything into two parts requires no thinking, while dividing by three or four is already pretty complicated.
Most currenct currency systems have coins and banknotes with values such as 1, 2, 5, 10, 20, 50, 100, 200, 500, 1000. Some currency systems do not have 2, 20 and 200, but almost all have those beginning in 1 and 5. At the same time, most currency systems do not have a coin or banknote that begins in 3, 4, 6, 7, 8 or 9. Interesting, isn't it? But why is that so?
Because you always need either ten of the 1s or two of the 5s (or five of the 2s) to arrive at the next bigger order. Calculating with money is very simple: times ten, or double. Just two kinds of operations. Every coin that you have is either half or a tenth of the next order coin. Those numbers multiply and add up easily and well.
So the 1 and 5 have been deeply ingrained, from their earliest childhood on, into Fisher and whoever else chose the significance levels as the most straightforward, most simple, most basic divisions of 10. Any other number needs an argument for it, while these numbers are simply there.
In the absence of an objective way to calculate the appropriate significance level for every individual data set, the one and five just feel right.