Most of the reasons have already been mentioned, but for me tire choice is for a few reasons:
1) to have a tire tough enough to handle the terrain. Travelling fast (60+mph) down a rough rocky road is no big deal for an off-road tire especially when it's not carrying nearly the same load as a truck. A trailer tire or other passenger car type tire would be far more susceptible to punctures in those conditions. Also on the more technical trails the same principle applies when there is risk for sidewall punctures in difficult rockcrawling type trails. Off-road tires have more durable sidewalls.
2) Have tires that can be used on the tow vehicle in a pinch is nice. Usually you won't find yourself in the situation where you need to canabalize tires from the trailer. But the first time you are in that situation, you'll kick yourself when you have to walk out because your tow vehicle is disabled, and you could have avoided that by having tires on the trailer that will fit the truck.
3) For me it helped get the trailer to the correct ride height. You could build a trailer at the correct height using trailer tires, but I went the other direction.
4) Poor man's suspension. The off-road tires on my trailer are designed for much heavier load than I ever have in my trailer. And since my budget trailer uses standard 3500# trailer springs, the suspension doesn't move a lot on it's own. I generally run the tires on trailer at 10-12psi. It has proven to be more than adequate for the load of the trailer and I've never noticed the tires getting hot from underinflation. And with the stiff springs, the low air pressure helps smooth out the bumps. Remember that comment about 60+mph down rough roads? Yeah, ask me how that works with 25-35psi in the trailer tires.
That speed over rough roads at 10psi is surprisingly nice. A more sophisticated suspension would be more ideal, but not everyone can afford that. I couldn't and preferred the simplicity of the leaf springs over air bags and trailing arms with more components to fail.
5) Looks. Hard to deny that the trailer looks cool with big off-road tires. I'm vain, so sue me.