What made it clear for me was this : Since there are 2 empty boxes, this means that if you pick some box out of 3, you will have 2/3 chances of picking an empt

What made it clear for me was this : Since there are 2 empty boxes, this means that if you pick some box out of 3, you will have 2/3 chances of picking an empty box. However, the revealed box will always be empty, so in both of these outcomes the revealed box will be the other empty one. So in 2/3 of cases, the pair (picked,revealed) will contain both of the empty boxes, and so switching will get you the prize