It means that you shouldn't change out of your winter woollies until May is over - i.e. you shouldn't be wearing your summer goodies unitl the 1st of June (technically the first day of Summer!) - but bearing in mind it is an old Scottish Saying - so we were always a little chillier there!! - and we used - 'ne're cast a clout till May be oot!'
It is supposed to mean that you shouldn't start wearing your summer clothes until the may blossoms on the tree- if you waited until the month of May was over you would be extremely warm!