How is installing roof sheathing horizontally stronger than vertically?
The same policy goes for roof sheathing as for hanging drywall, you get more strength by spanning more rafters/joists/studs and offsetting the joints. This solves two points of weakness.

The main one is the structure turning into a parallelogram where the studs are no longer perpendicular to the ground. When you think of using a diagonal brace to support a wall, you want to get both ends as far apart as possible. If you install vertical, then that gives you the support of a diagonal brace spanning 4' worth of studs. But by staggering the joints and going horizontal, you get the equivalent 12' or more of studs braced.

The other form of support you get is for vertical load on a roof segment. If you stand between two rafters and the sheathing was laid vertically, you are closer to a seam where the nails could pull away under load and you can fall through. Holmes on Homes had an episode where they were replacing a leaking roof and the builder laid a patch where the owner decided not to go with the skylight option. The patch was laid vertically, and they almost fell right through the roof.