Reciprocity essentially says you can swap the light source with the camera and still get the same result. This is important because it justifies the reversing of light paths in the path tracing algorithm.
I'm not sure how to intuitively understand the energy conservation inequality on the bottom. I think I see that it has to do with looking at pairs of vectors coming out/going into the center of a hemisphere and we put in cos theta because of Lambert's cosine law but I'm not sure how to piece this all together with the inequality.
I think it means that the energy never increases but always decreases for any outgoing direction. Its either absorbed into the material or some portion less than or equal to 1 is reflected back out.