The two-word answer to the question of why mountain biking improves visual acuity is: varied terrain.
All biking requires a heightened attention to your surroundings. On a road bike, the consequences of a failure to monitor your environment can be very serious indeed. The same is true on a mountain bike, but when navigating off-road the terrain you are riding over tends to vary almost infinitely. Stones, tree roots and water hazards can appear out of nowhere when you are traversing a mountain trail. Branches, stumps and that steep uphill portion of the trail ahead require that you engage in a constant visual exam in order to maintain your balance and forward momentum.
The challenge of varied terrain is part of the fun of mountain biking; and your eyes provide most of the information that your brain requires to keep you upright and pedaling safely on a mountain bike.
As anyone who rides a mountain bike knows, you receive a full workout of mind and body during a good, long mountain bike ride. Swimming and cross-country skiing may provide a more complete overall workout than mountain biking, but not by much.
The eye-hand coordination required by mountain biking is unsurpassed.