I would wear a nice pair of Khakis and a polo. Not anything over-dressed but certainly not underdressed. No heels, nothing open toed, preferably something 'barn worth'. By pasture boots do you mean paddock boots? If so, I think those would be appropriate.
Everyone has given you great advice on how to interview. Heres a few more points.
Be on time. I know that is a given but it is essential.
Don't ogle the horses. We all love horses and we all want to fuss over them. Try your best to not look like a kid in a candy store. Compliment them on a nice barn but I would refrain from commenting on the horses at all.
Always thank them for inviting you out to interview and shake their hand.
If you do have references, have them typed neatly and ready to hand over.
Good Luck! It sure looks like a nice place!