I agree with Nathaniel's answer, but I'd like to offer a different way of looking at this which I would claim is both more historically natural and easier to understand because it's more concrete.
You, as Einstein's reader, are presumed to have read the first 1905 paper and to have understood its material about kinematics, electromagnetism, and the kinetic energy of the electron (not of a macroscopic body). But you don't know anything else about relativistic mass, energy, or momentum, and you don't know anything about how these quantities transform except in the nonrelativistic limit. Let's consider two examples of types of energy.
You know about potential energy, and you know that it depends on the distance between objects. Einstein describes a body, such as a lantern, that emits rays of light in opposite directions. If you imagine that the lantern contains particles separated by various distances and interacting through various forces, then it contains some amount of potential energy $U$, which depends on the distances between the particles. Nonrelativistically, $U$ is frame-independent, but you know from the first 1905 paper that relativistically the distances between the particles will contract when we transform from the $(x,y,z)$ rest frame to the new frame $(\xi,\eta,\zeta)$. If you express $U$ relative to its value when the lantern is disassembled into atoms, then presumably $|U|$ will increase because of the decreased distance. You have no way of calculating this increase, but you expect it to exist.
From reading section 10 of the earlier paper, you also know the relativistic expression for the kinetic energy of an electron, and you know that it differs from the nonrelativistic expression $(1/2)mv^2$. An object such as a lantern contains some amount of thermal energy $E_T$. In nonrelativistic mechanics, you know that there's a theorem stating that if a system of particles conserves energy in one frame, then you can infer that it would conserve energy in another frame if and only if momentum is also conserved. This theorem vastly simplifies classical mechanics by, e.g., allowing us to ignore the thermal energy of objects in elastic collisions. But the theorem assumes the exact $v^2$ dependence of kinetic energy on velocity. The relativistic dependence isn't $v^2$. The lantern is made out of some materials that have some microscopic structure, containing some amount of thermal energy $E_T$. This $E_T$ is going to transform in some complicated, nonlinear way because of the relativistic transformation of the microscopic kinetic energies. You have no idea how to calculate the transformation exactly, but you expect that because the relativistic expression for kinetic energy grows faster than $v^2$ at large velocities, the lantern in the $(\xi,\eta,\zeta)$ will probably have some excess of thermal energy relative to what you would have expected nonrelativisticvally.
When Einstein constructs a difference like $H_0-E_0$, you therefore expect a bunch of different terms to contribute to it physically. At a minimum, you expect it to contain:
the macroscopic kinetic energy $K_0$ of the lantern
an anomalous relativistic effect on the potential energy $U$
an anomalous relativistic effect on the thermal energy $E_T$
The constant $C$ is meant to absorb effects like 2 and 3.
With modern hindsight, we know that mass-energy is the timelike part of the energy-momentum four-vector, and therefore all of these effects can be calculated in a uniform and simple way. As Nathaniel says, the transformation from the rest frame $(x,y,z)$ to the $(\xi,\eta,\zeta)$ frame simply multiplies the lantern's mass-energy by a factor of $\gamma$. But Einstein hasn't established that yet, and that's why he can't assume anything about $C$ other than that it's a constant that "doesn't change during the emission of the light."