Below your Camera's X-sync limit (often around 1/250, what your question refers to as "normal conditions"; any of the Modes (Flash, HSS, HS) is "normal" for some people, so call it X-Sync Limit) a simplified explanation is: The Shutter opens, the Flash fires, the Shutter closes. So the Flash is faster than the Shutter and thus the Shutter doesn't control the Flash's intensity.
Above the X-Sync Limit the Shutter is so fast that the Second Curtain (the closing half of the Shutter) must start to move before the First Curtain (the opening half of the Shutter) has moved; resulting in a moving Slit that travels across the Film or Sensor.
The HSS Flash is pulsed repeatedly to provide continuous (as well as diminishing, and differing color) light for the short duration of the two Shutters (moving Slit) AND a higher Shutter Speed makes the Slit narrower cutting the Flash's power so much that the Flash MUST be fairly close to the subject (when shooting into the Sun the Flash is almost in the Shot.
With HS (the third Mode, which you asked nothing about) the Flash fires a single time, slightly before the Shutter with an enormous pulse (distances of several hundred feet away from the Flash can brightly light the subject) and the electronics times the Shutter to utilize the hottest portion of the Flash. So you're (sort of) back to a normal Flash (not HSS).
HS is what you want to buy/learn IF you're into using Flash.
HSS was developed to use a Flash at higher Shutter Speeds without getting a black bar in the Picture (due to the Curtains blocking part of the Flash) and eats your Battery - HS only eats as much Battery as the brightness to which it is set. Greater investment but more Shots per Battery (for the intensity used).
That was supposed to be an oversimplified (but not incorrect) explanation.
Marc Weiler's Website features many Photos taken with a Elinchrom Hi-Sync, in particular Photo number 13 on this Webpage shows the use of a Flash at a very long distance and a clear view of the amount of light it casts at that distance. Technical Note: You can click on one of those Photos and they open in a Gallery if you don't wish to count.