Because he does things like this when his colleagues leave for greener pastures.
Jon Oliver — who famously took over the desk this summer — is getting his own show on HBO. Stewart is clearly proud and happy, and wanted to give him a bit of a send-off on-air. He also wanted it to be a surprise, though, so he wrote and rehearsed a completely false bit that required Oliver as the correspondent, and then broke character about a minute into it to shift into a really glorious sendoff compilation.
It seems pretty clear that Stewart’s a swell guy, and that the team at TDS have real affection for one another. It shows through the comedy, and probably fuels their success to no small degree.