We know he won't be executed on Friday. Assume he'll be executed on Thursday. Then it means that by Wednesday, he'll know that he'll be executed the next day.

We know he won't be executed on Friday. Assume he'll be executed on Thursday. Then it means that by Wednesday, he'll know that he'll be executed the next day. So it won't be a surprise then. But this contradicts the premise that the day of execution will be a surprise. So the assumption that he'll be executed on Thursday leads to a contradiction. Hence he won't be executed on Thursday. Similarly, Wednesday can be eliminated, and then Tuesday, and then Monday. Which contradicts the premises. But the premises were true anyway.