First, SO(3) of course has its unique invariant probabilistic measure. Hence, “random rotation” is a well-defined SO(3)-valued random variable. Each rotation (an element of SO(3)) has an uniquely defined rotation angle θ, from 0 to 180° (π) because of axis–angle representation. (Note that axis is undefined for θ = 0 and has two possible values for θ = 180°, but θ itself has no ambiguity.) Hence, “angle of a random rotation” is a well-defined random angle.
Why is its average closer to one end (180°) than to another (0)? In short, because there are many 180° rotations, whereas rotation by zero angle is unique (identity map).
Note that I ignore Spin(3) → SO(3) covering that is important in quaternionic discourse, but it won’t change the result: Haar measure on Spin(3) projected onto SO(3) gives the same Haar measure on SO(3), hence there is no difference whether do we make computations on S3 of unit quaternions (the same as Spin(3)) or directly on SO(3).