Either you can say the second postulate is an experimental fact - people measure light's speed in different frames directly or indirectly and get that the speed of light is the same in all the frames. Or, you can derive this fact theoretically from Maxwell's equations with a little aid of the first postulate.
Maxwell's equations read as following in vacuum:
$$\nabla \cdot \vec{E} = 0$$
$$\nabla \times \vec{E} = -\dfrac{\partial \vec{B}}{\partial t}$$
$$\nabla \cdot \vec{B} = 0$$
$$\nabla \times \vec{B} = \epsilon_0 \mu_0\dfrac{\partial \vec{E}}{\partial t}$$
Therefore, $$\nabla \times (\nabla \times \vec{B}) = \epsilon_0 \mu_0 \dfrac{\partial}{\partial t}(\nabla \times \vec{E}) $$
Or, $$\nabla^2\vec{B} = \epsilon_0\mu_0\dfrac{\partial^2\vec{B}}{\partial t^2} \tag{1}$$
Similarly, $$\nabla \times (\nabla \times \vec{E}) = -\dfrac{\partial}{\partial t}(\nabla \times \vec{B}) $$
Or, $$\nabla^2\vec{E} = \epsilon_0\mu_0\dfrac{\partial^2\vec{E}}{\partial t^2} \tag{2}$$
Equations $(1), (2)$ are simply the equations of waves of $\vec{E}$ and $\vec{B}$ propagating with a speed $\sqrt{\dfrac{1}{\epsilon_0\mu_0}}$ - which is completely the same in every frame the Maxwell's equations are valid. And now, from the first postulate, we can say they are valid in all the inertial frames (or you can call it an experimental fact if you wish). This is how we conclude that the speed of electromagnetic waves (which is light) is the same in all the inertial frames.
But we call it a postulate in the sense of postulating that Maxwell's equations are true laws of Physics and are, thus, according to the first postulate are valid in all the inertial frames.
Edit Regarding the relation of the second postulate with masslessness of particles: As such the second postulate doesn't refer to any particle at all. It (as illustrated above in this answer) refers to waves and their speed of propagation in the vacuum. So, in order to get to the second postulate, we don't need to know anything about massless particles. But, once we have these postulates, laws of momentum conservation and energy conservation forces upon us that the energy of a particle must be $\dfrac{E_0}{c^2\sqrt{1-\dfrac{v^2}{c^2}}}$ and that its momentum must be $\dfrac{E_0 v}{c^2\sqrt{1-\dfrac{v^2}{c^2}}}$ (where $v$ is the speed of the particle and and $E_0$ is the energy of the particle in its rest frame). From these formulae, it becomes clear if $v=c$ then the energy and momentum would become infinitely large unless $E_0=0$. This is to say that the only possible way a particle can travel at the speed of light is for the particle to be massless (i.e., $E_0 = 0$). On the other hand, the only way for a massless particle to have finite energy and momentum is to travel at the speed of light. So, if a particle is massless then it has to go at the speed of light as well as if a particle goes at the speed of light then it has to be massless. But this is something we derive from Special Relativity - not something we postulate to derive Special Relativity.
