* Free, comfortable movement with no rubbing and chafing, and, in case of full-seat breaches - added stickability, if needed.
* Very comfortable to wear under chaps or long boots.
* Prevention of catching yourself in the tack, prevention of getting pinched by the stirrup leathers;
* In training, the trainer can observe much easily everything that's going on with the riders' legs in nuanced details.
And the people who look stylish in breeches are really the minority!
I love my breeches. I've got three pairs of them. Not that I can't ride in other clothing, it's just the most comfortable option I've found. ;)
Last, but not least, wearing clean and proper riding attire is a sort of tradition and a sign of respect towards your trainer in many classical riding schools in Europe. You may wear whatever you prefer at home, be it a kilt, or a pair of worn and muddy jeans on a rainy day, if you please, but out of respect of the equestrian world one is a part of, it is expected to be able to follow the tradition. It's just like everyone understands that wearing a pair of muddy jeans is not acceptable if you're going to an official business meeting or something like that.