One of the greatest things about the guitar is it's ability to be quickly and easily tuned to whatever you need it to be tuned to to make the song you're singing sound amazing! (DGdgbe for slide DAdgad for celtic CGdgbd slack key for Hawaiian slide... pretty close to the one he's using.) I think the guitarist in the video did a great job of matching the tuning and the song.
In answer to your questions...
Why would you tune your guitar this way?...to make the song sound awesome on guitar.
Would it make it easier to play certain chords?... Yes in the case of slide tunings all you have to do is barr the chord. In the case of celtic it gives you the ability to have awesome drone sounds.
Is is just to create a trademark, stylistic sound?... No you'd be pretty limited if that were the goal. Case in point goo goo dolls, sonic youth, Leo Kottke...it's a part of the sound, but not THE sound.
Would it make any chords any harder to play?... Yes in modal tunings thing get offset and shuffled around quite a bit.
Is it just to show off his knowledge of music theory? I'd be willing to bet that it's just a tuning the artist likes to play around in.
Someone online alluded to it being "basically Open D with the 6th string lowered a whole step" but when I googled "Open D tuning", it looked nothing like it... I'm thinking it's more of a modified slide tuning.
Alternate tunings are just one of the many parts of the guitar as an instrument that makes it such a perfect animal for expressing your creativity.
If you want to learn more about alternate tuning in general, check out Mark Hanson's book...The complete book of Alternate tunings.
Peace