The photons produced in nuclear fusion reactions travel less than a mm before they are absorbed inside the Sun. They don't "fight their way out". Instead there is a gradual diffusion of radiation outwards because of the temperature gradient that exists between the inside and outside. As that temperature decreases, the dominant photon wavelengths in the (approximately) blackbody radiation field get longer and longer, but individual photons do not escape the Sun.
The photons we do receive at the Earth are mostly visible and infrared photons produced in the photosphere, which is where the Sun's atmosphere finally becomes thin enough and cool enough to be transparent to the photons produced there. Hence they escape and may travel to the Earth.
As to what exactly produces that visible light, there are a variety of mechanisms. However, the dominant one at most wavelengths through the visible and near infrared regimes is the photo-recombination continuum radiation (free-bound radiation) produced when free electrons (made available by ionising sodium and potassium atoms) attach themselves to hydrogen atoms to form H$^{-}$ ions.