As was said, the mechanical shutter has speed limitations. As to the slit, try to imagine without it. Suppose the shutter opens by moving from top to bottom of frame. And then of course, it has to close from bottom to top. So it is open longer on the top side than on the bottom side, which is uneven exposure.

Modern fast curtains might move about 7 meters per second, but still, the time to cover 16 or 24 mm of frame height is a factor for a fast shutter.

Crossing the 24 mm frame height at 7000 mm per second takes about 1/300 second to completely open. And if another 1/300 second to close, then fastest shutter exposure might be near zero at the bottom edge of the frame, but maybe 2/300 or 1/150 second of exposure at the top edge. That much longer at any shutter speed, and of course, the shutter speed cannot be faster then the 1/300 second travel time to open. This is mighty poor photographic performance.

Leaf shutters do try to improve this speed (with the multiple iris blades), but they still have to open and close, and they can't do this well at fastest shutter speeds. We might see 1/500 second on them, but not 1/8000 second on a leaf shutter. But the focal plane can do it, and accurately too.

But imagine a slit... When instead, if (as is done by focal plane shutters), if one curtain moves across opening the frame, and then a second curtain follows by moving across closing that frame, then all we have to measure is the timing between starting each curtain. Both frame edges get exactly the same exposure (but occurring at very slightly different times). And the shutter speed can be very fast, say 1/8000 second, which of course has the effect of being a narrow slit moving across the frame, but it is very even exposure, and very precisely timed. Mechanical gears and springs and motion are no longer factors, since all motion is exactly the same for any choice of shutter speed, and the same for both curtains. No mechanical variations due to various shutter speeds. We can simply use a quartz clock crystal to measure the two start curtain times, which difference becomes the shutter speed duration. It does not get much better than that. Even exposure and precise.

However, this becomes a problem for flash. Flash is very fast (a short duration) and can only usefully happen if the shutter is fully open to pass it. So that means a slit cannot work for flash, or rather, the slit has to be the fully open sensor dimension. It means that this 1/300 second travel time to open is the shortest shutter duration that can work for flash, called shutter sync speed.

A 1/200 second sync is more common in lesser cameras, but the fastest focal plane shutters can and do 1/320 second (several Nikon DSLR models do it).

If it takes 1/300 second to fully open, and if the second curtain starts closing about immediately then, this still leaves a tiny fully open window for the fast flash to occur. That minimum shutter speed for flash sync is necessarily 1/300 second. Flash is a minor downside, but it really is all rather ingenious. And the focal plane shutter idea is about 100 years old.