The only reason I could see for asking a horse to lay down is for disabled folks who can't mount properly or easily. Other than that I dunno... But I have met quite a few who were trained to lay down for this reason.
If a person is too disabled to mount they are too disabled to deal with the motion of a horse standing up. This is why we use mounting ramps, so the chair can bring them level to the horse, so it's just a matter of pivoting them over to the horse.
As for the OP, laying down there are 2 ways to teach this skill, for 2 different reasons:
1) the horrifying act of laying a horse down for submission - I personally hate this, but I've seen it make a world of difference in some exceptionally violent horse's lives. It's forceably twisting a horse off it's feet and onto the ground, and holding them in submission. When they get up, trust me, they are submissive.
2) for fun! It's a fun skill to teach, my favorite way to teach the skill is with clicker training. Assuming the horse has appropriate clicker training foundation and has learned how to take treats and has learned the click bridge. You can spray them with a hose, then when they go to roll, click and treat when they lay down. Do this a few times until the behavior is learned. Make sure there's a clear cue for it - you don't want your horse thinking he can just lay down whenever!!! :P
It is a clear sign of trust - I would never force a horse down unless a life or death situation - but it's a fun skill to teach if you can do it right. :)