The sun appears 13x larger because the suns corona, that extends for 12 sun radii out from the surface of the sun, is visible as a luminous disc. The same form of "light pollution" that prevents you from seeings stars in cities, also prevents you from seeing the suns corona along the ecliptic. Along the ecliptic there are dust particles that orbit the sun on the ecliptic plane just like every planet in our solar system does. These reflect light, and can be seen for example with the "zodiacal lights". The Voyager 1 probe was looking at the solar system from above, at a 32° angle to the ecliptic.

The position of Venus and Earth relative to the sun, makes it easy to make out that the size of the object seen in the Voyager 1 image is the same as the suns corona, the distance from the sun to Venus is around 5x the width of that object, and that fits perfectly with a solar corona 13x the diameter of the sun (NASA, 2014).

During day time, if you measure the brightness of stars, you get no reading at all, since you are measuring the contrast between the star and everything else, and there is a lot of "light pollution". The same is true for measuring the brightness of the corona, just like the Earth's atmosphere prevents you from seeing stars in the day, the dust throughout the ecliptic also lights up like an atmosphere, making any measures of the brightness of the corona appear low relative to the overall brightness of the region of space you are in.

The Voyager 1 image also shows glare rays and lens flares, the hypothesis that the suns corona, which is luminous as you see in a solar eclipse for example, appears with less light pollution as a bright disc, does not reject optics in any way.

Synapses

1. NASA's STEREO Maps Much Larger Solar Atmosphere Than Previously Observed - nasa.gov

2. Solar System Portrait - View of the Sun, Earth and Venus - jpl.nasa.gov