Pranjal Singh, MechE (4th Year) at IIT, Delhi
Answered Jun 18, 2015 ·
All the answers here said most of it in terms of solid mechanics theory of pure bending of beams. Since you used the term 'shaft' meaning we're talking about torsion, I'll just add a layman explanation regarding that:

Since the mass is constant, A hollow shaft is necessarily a result of reallocation of material from axis to periphery. This results in decrease of slenderness of the shaft and hence, it's torsional rigidity increases.

Explanation:
The strength of a shaft implies the maximum torque it can transmit. It depends on Torsional rigidity of the shaft since it renders the stiffness to the shaft. Torsional rigidity is defined as the torque required to produce a twist of one radian per unit length of the shaft. and by torsional formula:

T/(θ/L)=GJ=Torsional rigidity.
So, Stiffness and hence strength is proportional to J which is:
by maths, less for solid shaft and
by intuition, less for most slender of the two (solid shaft).