The Doctor goes silent, unable to look Clara in the eye. His gaze is directed downward, at their feet. He stubbornly wants to resist, but realizes how much he's missed the comfort of physical contact, even something as simple as holding hands. It really hasn't happened since River Song had disappeared. He also realizes that, just like River, it's nearly impossible to say 'no' to Clara.
Remaining stoic, he agrees. They sit back down. Throughout the show, he remains silent, except for the times when he huffs out of embarrassment or amusement. If Clara asks questions, the Doctor answers quickly so they can go back to watching.
When it gets to the point of saying goodbye to "Bill" and "Nardole", the Doctor is choked up. But it's when "Clara" shows up, he's absolutely stunned, and a tear runs down his face.