The actual physics are a lot more complicated than my summary but this is what basically happened.
There were no photons during the Big Bang. Particles such as photons happened after the universe had already started expanding and cooling. Around age 1 second, neutrons and protons had formed. This process is still not observable even with the strongest telescopes because the universe was opaque. Light was continuously absorbed and re-emitted by countless sub-atomic particles. Technically this is called Thomson scattering where photons only travel a small mean distance before encountering a free electron.
After about 380,000 years, the first atoms were stable enough to form. "Space" became optically clear as protons and electrons combined into neutral hydrogen atoms and light could move. This first light is actually the cosmic microwave background radiation and we can and do see it with our instruments. There are some very detailed maps of the radiation created over the years.
For a better layman's explanation, look up "recombination (cosmology)" on Wikipedia.