Special relativity does not explain the expansion of the universe, general relativity does. It is a consequence of general relativity that any universe with a mass(energy) density below a certain threshold (the critical density) will expand. A universe above the critical density will collapse again. Observation shows that the density of our universe is slightly below the critical density which means it will be expanding forever.

This does neither interfere with the speed of light being constant and the same everywhere in a vaccum and nothing but massless particles being able to move at that speed. A photon still travels with c, however, the distance it has to travel is stretched over time and thus is the wavelength of the photon, what is called the cosmological redshift. Expansion may also cause objects to seperate from each other with a speed greater than the speed of light if their seperation is big enough already. They become physically unable to interact with each other then. For all practical purposes they are in different universes. This defines a universe with us in the center and a radius beyond which no other physical object is ever able to interact with us (i.e. we will never see light from that object arriving at our place). Given that expansion is accelerating this radius will decrease over time.