They never go in to it, not even a little bit. Just say that what the problem is, and the only way to fix it is to deliever this thingie to it, end of story. In fact, it is very much bright throu out the movie, burning several things to a crisp.
However, I can take a stab at explaining that one. The sun doesnt have to be dying per se, rather, no emit enough energy to heat up our planet. The sun itself is a relativly cool 6,000k, while the corona is close to a million degrees. So it would stand to reason that most of the light and radiation we receive from the sun actually comes from the corona. The corona is heated up by sounds, pressure and magnetic waves coming from the sun's core. When those waves reach the surface they release their energy in the form of light.
The sun itself orbits around the galactic core, traveling thru interstellas space at 200km/s. If it was to travel thru a region in which something were to dampen those waves only slightly, the corona would cool, releasing less energy to the earth. It wouldnt require a gread deal of cooling to seriously screw up life on earth.
Perhaps someone that knows their physics better can come up with something even more believable.