After months of wondering, at long the answer to the Breath of the Wild conundrum has made a splash online prior to an official reveal and all they do is pretty much the same thing amiibos did in Hyrule Warriors, they give you stuff Sure the give rare items like the Zelda amiibo gives the player a special shield.
Rider Link amiibo can give Link a large sword or a saddle for his steed, but the items are not amiibo exclusives. They can be found in the game, but amiibo just give them to you sooner. As for the other amiibo, Archer Link of course gives you a special bow and the Bokoblin amiibo can see you equipped with rather interesting looking clubs and items and the Guardian amiibo does the same, except it can give you “anti-guardian arrows.”
A GameXplain video can be found below for more information:
Source: GameXplain (YouTube)