Typical DSLR shutters are focal-plane shutters, which mean they open and close directly above the image sensor. They are mechanically primed, and when triggered, the "first curtain" opens across the sensor, followed by the second curtain after a brief delay corresponding to the relevant exposure time.
In the case of very fast shutter speeds (depending on the camera, typically 1/250s or faster), the second curtain begins its traversal of the sensor before the first curtain is fully open. This means that every portion of the sensor is exposed for the appropriate amount of time, but that not every portion of the sensor is exposed at the exact same time. This is due to mechanical limitations, and is hard to get around.
In general, this isn't an issue. However, flashes and studio strobes are of a very short duration - often 1/10000th of a second or less. In the case of a focal plane shutter, this can mean that not the entire sensor will be exposed to the flash's light, causing a stripe in the final image.
There have been various ways to compensate for this - for example, high-speed flash synchronization that fires the flash at lower power levels several times during the exposure such that each portion of the image sensor is properly exposed.
In earlier DSLRs (Nikon D50 and D70 models with the 6MP Sony CCD), a "digital shutter" was used, which allowed for native flash syncing at high speeds - the mechanical shutter only operated up to its "sync limit" of 1/250, and the digital shutter took over at speeds higher than that.
In comparison, leaf shutters always open fully, which means that there's a period where, if a flash is fired, the entirety of the sensor will be exposed evenly to that flash. Because of the very short duration of the flash, shutter speed does not affect how much of the flash's light falls on the sensor, and can be used to control for background light instead, as compared to aperture and ISO controls that would affect both background light and the impact of the flash's light in the overall image.