This post in the Hypnogoria blog suggests the following:
"Now the reasons for these strictures and taboos on changing bedding and garments in the month of May we can trace to a different origin than ancient Roman festivals of the dead; indeed it is perhaps the origin of much of the other lore of ill luck in May too. And like many things in the British Isles, it comes down to the weather! The clearest indication, and indeed explanation comes from an 1852 publication, The Englishwoman's Domestic Magazine Almanack, where it is written -
We warn young persons during this month not to throw off their warm clothing too suddenly, as in this changeable climate we often have a day of sunshine followed by a day of rain and hail. Many are the deaths by consumption, the seeds of which have been sown by this pernicious practice.
And there we have it - from sickly kids and kittens, to keeping cosy blankets and coats - all these forms of ill luck in the fifth month we can attribute to the changeable weather in May.
So, washing blankets refers to preparing to pack away winter bedding. The problem with doing that in May is that there's still occasional wintery weather during this month, so without proper warm bedding a person might catch a cold or worse and die.