In response to a Quora question, I wrote the following:
In his September, 1905 paper entitled Does the inertia of a body depend upon its energy-content? (Ist die Trägheit eines Körpers von seinem Energiegehalt abhängig?, Annalen der Physik. 18:639, 1905) Einstein investigates the energy $l$ (in the notation used in the 1923 English publication of the paper) of a system of plane electromagnetic waves, as measured by one observer. Another observer, moving relative to the first with velocity $v$ directed at an angle $\phi$ relative to the direction of the waves, sees the energy
$$l^*=l\frac{1-\frac{v}{c}\cos\phi}{\sqrt{1-v^2/c^2}}.$$
He then investigates a body that sends out two light signals with energy $\frac{1}{2}L$ in opposite directions (such that its momentum does not change). If the body's energy is $E_0$ before and $E_1$ after the emission, we have (due to energy conservation)
$$E_0=E_1+\frac{1}{2}L+\frac{1}{2}L.$$
In the other reference frame, let the body's energy be, before and after, $H_0$ and $H_1$. Then
$$H_0=H_1+\frac{1}{2}L\frac{1-\frac{v}{c}\cos\phi}{\sqrt{1-v^2/c^2}}+\frac{1}{2}L\frac{1+\frac{v}{c}\cos\phi}{\sqrt{1-v^2/c^2}},\tag*{}$$
or, after simple algebra,
$$H_0=H_1+\frac{L}{\sqrt{1-v^2/c^2}}.$$
Subtracting the two yields
$$(H_0-E_0)-(H_1-E_1)=L\left\{\frac{1}{\sqrt{1-v^2/c^2}}-1\right\}.$$
Now Einstein notes that $H_0$ and $E_0$ refer to the energy of the same body in the same state, in two different inertial systems; same goes for $H_1$ and $E_1$. On the other hand $H-E$ is the difference in kinetic energy, as seen in two systems that move relative to each other, up to some additive constant $C$ that is really just a matter of how the kinetic energy is defined in the two systems:
$$\begin{align*}H_0-E_0=K_0+C,\\
H_1-E_1=K_1+C.\end{align*}$$
So then,
$$K_0-K_1=L\left\{\frac{1}{\sqrt{1-v^2/c^2}}-1\right\}.$$
When the velocity is small, the square root in the denominator can be series expanded, and terms that contain higher powers of $v$ can be dropped, which leaves
$$K_0-K_1=\frac{1}{2}\frac{L}{c^2}v^2.$$
Note that the body's velocity does not change, yet its kinetic energy (which is $\frac{1}{2}mv^2$ when the velocity is small) changed by this amount after the emission. From this Einstein concludes that if a body gives off the energy $L$ in the form of radiation, its mass diminishes by $L/c^2$. He also notes that the fact that the energy withdrawn is in the form of radiation makes no difference, which leads to the conclusion that the mass of a body is a measure of its energy-content, with $c^2$ being the conversion factor between the two. (Note that the actual formula, $E=mc^2$, doesn’t actually appear in Einstein’s paper! But it is implied trivially by his result and is described in words.)
This was Einstein's fourth paper during his annus mirabilis (1905), and the second on relativity theory. Another example of an amazing early Einstein paper (see https://www.fourmilab.ch/etexts/einstein/E_mc2/e_mc2.pdf): short (less than three pages), no references, and so much to the point, it is almost impossible to summarize.
