There is a lot of truth to that and it can be explained with physics.

Assuming everything being equal (mainly the friction/adhesion force of the mud on the tire but this is a complicated function of rubber compound, tread design, mud wetness and even mud composition).

Assume the big wheel and the small wheel are traveling with the same linear speed, v, (such as 10 mph).

Mud can only stay on the tire if there is enough centripetal force, which is provided by the friction/adhesion force.

For the same linear speed, the mud on the smaller wheel needs to have a higher centripetal force to stay on because

Centripetal force = m * v^2 / R, where

m is the mass of the mud (same for both big and small wheel),
v is the linear speed (same for both big and small wheel),
R is the radius of the tire (smaller for the small wheel).

So the requirement for the centripetal force is greater for the small wheel, the mud cannot "make the cut" so it flies off along the tangent.

So it is like you have $100 so you can only buy a Wal-Mart blue light special for $88. The LBS $980 deal will fly you off along the tangent. Not everyone will be riding a LBS bike.