What is Big Macintosh wearing? Why is his tail short?

It's a horse collar, used for pulling plows and other heavy loads.

A horse collar is a part of a horse harness device used to distribute load around a horse's neck and shoulders when pulling a wagon or plow. The collar often supports and pads a pair of curved metal or wood pieces, called hames, to which the traces of the harness are attached. The collar allows a horse to use its full strength when pulling, essentially allowing the horse to push forward with its hindquarters into the collar rather than to pull with its shoulders as it would be required to do if wearing a yoke or a breastcollar.

As for his tail being short, this is a practice known as "docking":

Some horses used for driving still have the tail cut especially short to keep it from being tangled in the harness. In these cases, the term "docked" or "docking" only refers to the practice of cutting the hair of the tail skirt very short, just past the end of the natural dock of the tail. Though less drastic than amputation, it can still be difficult for a horse with a very short tail to effectively swat flies.