The same reason that guys who get grilled by the cops with no proof still confess.
When you know you are busted, regardless on if they can prove it, most people will panic and then immediately dig themselves a hole before they can pull it together to recognize they could lie out of it.
I'm surprised he/other pols don't just say "it's fake" and ignore the story. Maybe in this case there was another witness.
The same reason that guys who get grilled by the cops with no proof still confess.
When you know you are busted, regardless on if they can prove it, most people will panic and then immediately dig themselves a hole before they can pull it together to recognize they could lie out of it.