Meaning of Einstein's equation $E=mc^2$?

To understand the answer you should have understood well a point. Energy is conserved, mass is not. Moreover, masses possess an energy content pictured by the equation you wrote and thus this part of energy must be taken into account when writing the energy conservation law.

For instance, suppose you have a particle at rest in your (inertial) reference frame with a mass $M>0$. It may happen that the particle spontaneously breaks into a pair of different particles with masses $m_1$ and $m_2$ respectively (when measured at rest with them, I am assuming that all particles I consider are massive), but these particles are no longer at rest with you. Consequently they have kinetic energies, $K_1$, $K_2$, depending on their velocities.

Well, classically one expects the mass is conserved and thus $$M = m_1+m_2\:.$$ Instead this equation is experimentally violated.

Energy however is conserved, but part of energy is associated with the involved masses in accordance with the identity $E=mc^2$. Summing up the correct conservation rule is: $$Mc^2 = (m_1c^2 + K_1) + (m_2c^2 + K_2)$$ where $Mc^2$ is the total energetic content before the transformation of the initial particle: the energy is completely due to the initial mass. After the transformation, there are two types of energy to sum for each particle, the one due to the mass of the new particles $m_1c^2$ and $m_2c^2$ and their kinetic energy $K_1$ and $K_2$.

You see that, as $K_1+K_2 \geq 0$ you have that $m_1+m_2 \leq M$, so that the total mass decreased in this process.

There are other different kinds of transformations (for instance the inverse reaction $m_1+m_2 \to M$) also involving different types of energy (e.g. chemical or thermodynamical) but the point is that

(a) there always is part of energy due to the (rest) mass of the involved bodies, and here the celebrated equation $E=mc^2$ enters the picture;

(b) the total energy (for isolated systems in inertial reference frames) is conserved in time in any transformation process.