Say your dSLR has a native sync speed of 1/125th then at any speed higher than 1/125th the camera is in fact exposing for 1/125th of a second, but is doing so with a slit only a fraction of the height of the frame, such that any part within the frame sees light for less time. At 1/1000th of a second for instance the slit is only 1/4th the height of the sensor.
If you need further illustration you can google around for "focal plane shutter" and "flash sync" and I'm sure you'll turn up dozens of illustrations and videos.
You can use a flash sync system like Canon's HSS or Nikons' equivalent which turns the flash from a single-flash into a defacto continuous light source. This allows you to sync even at 1/8000th, but at the cost of losing a lot of light compared to the standard maximum for that flash using traditional flash sync. Likewise there are systems that allow you to use studio strobes with very slow/long flash durations that will last throughout the total transit time of the shutter, thereby dragging the flash along the entire transit of the slit. This likewise loses power as a ratio of [stated shutter speed / actual transit time] as compared to a strobe with normal flash duration which fits within a standard exposure. It also creates an uneven distribution of light across the frame since strobes fire in a pulse which starts at nothing, ramps up to the peak output, and the falls back down; though for many strobe/shutter-speed combinations the extent of unevenness can be made to be negligible.
A Leaf shutter lens usually has a slower maximum shutter speed, but because of the radial/rotational manner in which the blades of the shutter open and close the entire frame is exposed at once no matter what speed it's set to. The Phase One and Leaf backs employ an additional trick of sensor timing and electronics to extend the max sync speed to 1/1600th from a mechanical 1/800th.
Leaf Shutter Lenses are a pretty niche market and always have been. For people who need them there are really very few alternatives that don't involve major compromises (e.g. shooting through several stops of ND filter on a dSLR or having to have several times the number of lights to make a HSS-or-similar sync option work). For people who don't need them there are very few advantages and obvious disadvantages (leaf shutters add weight, complexity-i.e.-cost, and size to any given lens design). The best system (in my highly biased opinion) is one like the Phase One DF platform which allows for both Leaf Shutter lenses and standard Focal Plane lenses.
I suspect the reason Canon/Nikon have not pursued it is the very high cost of developing an entire system of new lenses vs. the return for their market. Developing an entire lens lineup for the small % of the market that love/need/want leaf shutter lenses probably does not make sense for them. Especially when there are entrenched players in that market that already have leaf shutter lenses as well as a greater focus on the feature sets (very high res, very fast tethering, large sensors, waist level viewfinders, compatibility with tech/view cameras) most likely to be desired by those in the market for leaf shutter lenses.
At some point it's very likely that most pro cameras will have "global shutters" whereby the sensor itself simply flickers on and off for whatever desired length and thereby can sync without any physical shutter at all. However I predict (only an informed guess mind you) that this will be 3-5 years from now. You can already see the implications on some smaller point and shoot systems (the sensors for which this is much easier to implement).