If you shine a torch (flashlight) on a wall and walk forward, the circle of light gets smaller, but brighter at the same time. The principal of the speed booster is the same.
A lens designed for 35mm projects an circle of light at least 43mm in diameter onto the sensor. The sensor in an APS-C format camera has a 28mm diagonal. The "speed booster" concentrates this 43mm circle down to a 31mm circle. Because the same amount of light is now falling on a smaller area, the amount of light per unit area is now increased.
Looking at it from a different point of view, the speed booster reduces the focal length of the system whilst the physical size of the aperture opening remains constant. Thus the f-number, which is the focal length divided by the aperture diameter, decreases.
Focal length changes by a factor of 0.71, thus the f-number changes by a factor of 0.71 which just so happens to correspond to one stop.
Fundamentally it's the exact opposite of what happens with a 1.4x teleconverter. A teleconverter increases focal length whist keeping the aperture diameter constant. Or alternatively a teleconverter enlarges the image circle but reduces intensity at the same time.
There are some other bold claims made by the manufacturers of the speed booster. In addition to increasing speed by a stop they also claim the resultant image is sharper, which goes against conventional wisdom.
However, the statement "when you add elements to a lens, you make the image worse" is not true absolutely (obviously if you remove elements from a lens you can make the image substantially worse, thus it is possible in theory to improve the image quality by adding elements).
It's true that each extra glass element will increase internal reflections and potentially introduce aberrations. Most add on filters are designed to make the lens do something it wasn't designed to do, e.g. focus at macro distances. However it's possible to use additional elements to correct aberrations present in the original design.
The speed booster falls into this category, in addition to shrinking the field of view the adaptor corrects for aberrations due to film era lens designs not accounting for digital sensor filter stack. The speed booster also increases telecentrcity, i.e. it makes the light rays strike the sensor more head on reducing vignetting and cross talk issues.
So if the claims are true, why hasn't this been done before? Users of APS-C DSLRs have long sought faster wide angle lenses, and teleconverters have been very popular for years.
Well, the problem is that teleconverters increase the backfocus distance, i.e. they cause the focussed image to be projected further behind the lens than it would be without the teleconverter. This isn't a problem as the lens/converter can be moved further from the film plane using a simple tube.
Focal reducers on the other hand (Speed Booster is just the product name of a particular focal reducer made by MetaBones) cause the backfocus distance to get smaller. With a DSLR there simply isn't space for the adaptor, and any optics to increase backfocus would negate any gains in image quality.
However if you take a lens with enough backfocus to accommodate a full frame SLR mirror and mount a focal reducer there's just enough space left for use with a mirrorless camera.