Axel, the widening of the field of view allows one to move the camera closer to fill the frame and this reduces the depth of field.
The depth of field a lens and sensor combination can create is at it's maximum at minimum focus distance.
With smaller sensors that result in larger crop factors, like the GH4, a 50mm full frame lens gives a Fullframe field of view of 100mm.
Let's bring this into the real world of cinematography: if I use my Panasonic GH4 to shoot a close up dialogue scene with an adapted Sigma 35mm f/1.4 I get a field of view of 70mm, perfect for a proper 'head and shoulders' close up.
A 35mm even at f/1.4 is no bokeh monster, especially on a m43 sensor, so by using the Speed Booster on the 35mm f/1.4 I gain .71x in field of view and one stop of light, so now it's an effective 49.7mm f/1.0 instead of 70mm f/1.4.
Now I can lower my ISO one stop to maintain the exposure and lower noise without penalty AND I'll reframe by moving the camera closer, thus nearing the minimum focus distance. This results in a decrease of the depth of field.
If I didn't want to move the camera I could also use a Sigma 50 f/1.4, which would normally be a 100mm field of view, but with the Speed Booster, to becomes an effective 71mm f/1.0, extremely close to the previous 70mm effective view with the Sigma 35mm f/1.4 (without booster). But now I have increased the focal lengthy optically, thus decreasing the depth of field.
The Speed Booster opens up many avenues for low-budget film projects.
I can keep noise much lower by using optically fast lenses to reduce the ISO.
I can lower the power level from external lighting like LED panels and save battery life or reduce heat buildup from HMIs thanks to the faster apertures.
I can use incredibly fast, light gathering T-stops with no danger of approaching the ridiculously thin full frame depth of field that only results in most things annoyingly out of focus. Yet I can still enough beautifully blurred foregrounds and backgrounds.
I can regain and harness some of the lens character that's lost due to the 2x crop that only uses the 'sweet spot' center. Sometimes the lens edges add to the cinematic feel.
If the BMCC Speed Booster usage on the GH4 is worked out and deemed safe, all the above applies still and is improved upon.
These are exciting times to be in cinematography.