It's essentially a way of using image stitching so you can use a long lens to shoot a wide-angle image.
As with a panoramic image, you shoot a number of pictures that will eventually become one image. The idea is to use a relatively long lens, like a "portrait length" telephoto or longer, so that you can get a very shallow depth of field on your main subject. You then shoot a number of images of the subject's surroundings, keeping all of the settings -- including the focus point -- the same as the original image. This is a whole lot easier to do if everything, including focus, is manual.
Ordinarily, the primary subject is a person or a couple (Brenizer is a wedding photographer), and people move, their clothing shifts, etc. You want, then, to get a very tightly-framed picture of all (or at least most) of your main subject with the first shot. Sometimes you can cut them off at the feet and fill in the rest later, but if you're shooting people "dressed up" outdoors, the chances are that dress hems and trouser cuffs will go astray. Sometimes stitching makes up the difference, but not all of the time.
The next shots will be of the surrounding area. Make sure there's a generous amount of overlap between images, and take enough pictures to completely cover the same field of view as the wide-angle lens you are trying to simulate.
The rest is post-processing: panoramic stitching and whatever retouching you may need to do to make the image seem like it was all-of-a-piece in the first place.
When you are done, you will have created an impossible picture -- the equivalent, say, of shooting the image with a 24mm f/0.2 lens. That is, you'll get the field of view that a wide-angle lens will give you, but a razor-thin depth of field that a wide-angle lens can never give you. (Well, not any wide-angle lens that can be made with the materials and knowledge of optics that we're likely to have in the next century or so.)
The optimum for ease of shooting would be one or two people and a static environment. The more moving parts there are in the scene, the harder it gets -- people will have to hold a pose anyway, and the more people there are, the less likely it will be that everyone will hold still while they're being shot. Dogs and kids? Maybe not a good idea most of the time.