Arctan(friction coefficient) gets you the "friction angle", which is a meaningful angle to know in a lot of physical systems. For example, it tells you how steep of an angled surface your object can rest on before it starts sliding downhill.
In this case, it sounds like the friction angle is important because it's the angle at which the reaction force is being applied to the unicycle by the road. (The normal force and friction force components make a right triangle with the resultant force, and their ratio - the friction coefficient - is by definition the tangent of that angle). I know nothing specific about unicycles, but it sounds like they're saying the rider needs to lean back so that the "shaft" between the seat and the axle is oriented along that line of force or else the system will become unstable.
In this case, it sounds like the friction angle is important because it's the angle at which the reaction force is being applied to the unicycle by the road. (The normal force and friction force components make a right triangle with the resultant force, and their ratio - the friction coefficient - is by definition the tangent of that angle). I know nothing specific about unicycles, but it sounds like they're saying the rider needs to lean back so that the "shaft" between the seat and the axle is oriented along that line of force or else the system will become unstable.