How is energy from nuclear fusion transferred through the layers of the sun?

By radiation and convection.  

Step-by-step explanation:

Just outside the core is the radiative region.

Here, ions of hydrogen and helium emit photons that travel a short distance before being reabsorbed by other ions, and the process repeats many times.

After several hundred years, a photon reaches the convective zone, about 200 000 km below the surface.

The temperature near the radiative zone is about 2 000 000 K but only about 5700 K at the surface.

Pockets of hot plasma rise to the surface in convection currents, much like water boiling on a stove. This journey takes about 10 days.