The argument is the same for both photon and matter-couplings, but it is easier to think about for photons. If a particle has a photon coupling and has a lower mass than the energy of photons inside the Sun, then there is a chance that a photon can radiate one of those particles.

Have a look at the CAST experiment for one of the standard examples of searches for such particles.

For the layperson (and perhaps everyone), "new light degree of freedom" = new low-mass particle.