They stuck a bunch of motion capture balls on a horse and just recorded it as it ran on a treadmill at different speeds. That's why the movements are realistic.
The horses do look great as well. I'm really impressed by the graphics as a whole. With games this big, graphics sometimes suffer but R* pulled it off. Look at that rock face:
Plus the clouds and lighting is the best I've seen, personally. I read that the sun can effect how well you snipe in MP if the sun is behind you or infront of you. I've not seen evidence of it myself. I'm liking the sound of heavy rain causing puddles to form on the ground too.
Avoid any RDR site. The games leaked and streams and spoilers are everywhere.