This math is a bit over my head, but I'll hazard a guess as to how conservation of energy might lead to a proof that this series converges: the space of possible distributions of radiance throughout a scene is a normed vector space, with the norm being related to the integral of all radiances across the entire scene. Conservation of energy says that neither reflection nor light propagation can increase this integral; mathematically, this means the light transport operator has norm $|K| \leq 1$.
According to Wikipedia, the Neumann series of a operator with norm less than 1 over a complete vector space always converges. In a real scene, some radiance is necessarily absorbed and turned into heat (at the very least, the camera itself is absorbing light!), which gives us the strict inequality we need. I'm not sure about the second requirement, though -- I've never understood what it means in practice for a vector space to be complete.
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And just to make this all more complicated, radiance actually increases when passing through the boundary from a medium with a lower index of refraction and to one with a higher index of refraction! (Essentially, differential cones of rays are squeezed down into smaller differential cones; this cancels out when going the other way.)
This doesn't violate energy conservation, but it does complicate the analysis of the operator norms. See Chapter 4 of Eric Veach's thesis for more info...