To be honest, the 'why' in my case is mostly for its own sake (curiosity, looks, fun of building it). I've always loved 'Mad Max' vehicles that mix junk yard aesthetics with high tech parts and 'futuristic' design concepts.
In general, the possible benefits of using such a Hossack front end on a motorcycle (or trike, or sidehack rig) are primarily:
1) changes (hopefully improvements) to handling (mostly by eliminating brake dive common to telescopic forks, but also eliminating striction)
2) reducing stress on the frame (shorter lever arm between wheel and frame attachment)
I'm mostly interested in 1. Race bike builders (including Norman Hossack, the inventor of the system) take advantage of 2 to allow building a lighter motorcycle (both sprung and unsprung mass), but obviously that's not my goal (or really possible, given I'm adding it to an existing bike without otherwise changing the frame). A good example of a motorcycle where the system is used mainly to improve handling is the BMW K series bikes, with the Duolever system. My rear suspension setup is also similar to (older) BMW designs. In some ways, its a ghetto K bike.
What you likely see on trikes is a girder fork- I've only ever seen one example with a Hossack setup. A girder and a Hossack front look superficially similar (especially in my case, where the build isn't done yet) and both allow the same anti-dive braking benefits. That's important to trike builders because trike is much heavier than a motorcycle. Being so heavy, trikes have much worse problems with front end brake dive if using a conventional motorcycle fork. By building a Hossack system instead of a girder, I can also choose any rake angle I like without having to actually cut the frame and move the head tube, and tuned the wheel path / bump response so that rake and trail increase slightly with bump (using a telescopic fork, they decrease). I'm going for a rake angle of 20 degrees (or less), and also reducing the wheelbase a bit. Steeper rake angles are safe, as long as you maintain the trail at a reasonable figure (3-4"), and actually improves handling on rough surfaces. The Hossack design lends itself very easily to steep rake angles, compared to a conventional headset steered fork.