For most of my life I thought it was silly and pompous, especially with the English and their silly wigs, but recently I realized it kinda makes sense. Judges are supposed to be impartial and represent Justice. If judges wore their own clothes, it would appear they are biased in favor of the side dressing similarly or wearing the same symbols (i.e. a cross necklace) as them. The sides could also start imitating the judge's attire to appease them, all of which would distract from the case and its essence in favor of form. However, when judges wear robes they all look the same, yet different than anyone else, most importantly those they're judging, but they're still human. In the end, it appears as if Justice itself (with a human face) is judging you rather than John Doe.
I was interested in whether this was the reason why the robes were introduced, but every query of "why judges wear robes" brings up detailed explanations on when the robes were introduced and which laws proscribe them. Nothing about the reason for it is mentioned.
So, why do judges wear robes?
We can find evidence that standard robes were established in England by the time of Edward III (1327-77). The fabric for the robes was given to a High Court Judge by the Crown, certain colors and trims for certain seasons. The style of regular fashion during the 14th century might look a bit familiar as it is very similar to even a modern judges robes. This style was kept consistent, the colors changing with time. So, to begin with it wasn't so much a style that was created to separate judiciaries out, but a uniformity of colors and fabrics for an already existing fashion. In 1635 we have the Judges' Rules published stating a black robe faced with miniver (light-color fur) for winter and violet or scarlet robes faced with pink-shot silk taffeta for summer. This was only loosely enforced over the years the colors and fabrics varying by preference. There are now numerous rules depending on the country, most created during the late 19th or 20th centuries. The detailed history of judicial wear in the UK can actually be found on their website.