Blight is a fungus and comes from spores carried in the air. Your weather will have been warm, wet and humid giving perfect conditions for blight to form. When the blight inducing weather started you could have sprayed your potatoes with a copper fungicide but I don't particularly like doing this. You have possibly been unfortunate in being hit and you will find that your neighbours potatoes have not been affected.
If you are sure that it is blight then cut the shaws down to ground level and put these in the bin or burn them to get rid of the blight spores. Leave the potatoes until later this weekend or the begging of next week and lift them all inspecting each tuber for any sign of rot. You will need to inspect the tubers regularly and dispose of any showing signs of disease.

Also do not grow any vegetables from the solanum family in that area of soil for 7 years - the blight spores can survive for years.

To be sure it is blight, take a look at my photos. On page three, bottom row is two photos of blight, one on a tomato and the other on potatoes.