None of the other responses seem to answer the question "What is the name of the thing the solar filter is eliminating?"
In fact, the solar filter doesn't eliminate anything. It just makes everything a lot less bright.
The reason the white part of the sun looks so big is due to saturation of the film, CCD, or retina that you're using to look at it. There's a limit to the brightness that the film, CCD, or retina can register, and anything brighter than that is going to look white.
You see a lighted region around the sun due to scattering effects that others have mentioned, but it really is not that bright compared to the sun itself. When you use a solar filter, you can see that clearly. But with an unfiltered view, it is still bright enough to saturate the viewing medium, just like the sun does, and so it looks just as bright as the sun, even though it is nowhere close.