Here's a grossly oversimplified way to think about it:
Imagine a dartboard with several rings radiating out from the center. At each location in the result, a score is computed by putting the dartboard over the location and seeing where the vector points are on the dartboard. From that the score is tallied and the raster is made.
There are many variables to how this is computed:
-- the size of the dartboard (the kernel)
-- the shape of the dartboard (2D isometric or 'the same in every direction in x/y', i.e. a flat circle)
-- the way the dartboard assigns points (Gaussian implies a 'normal' distribution, i.e. higher scores as the point gets closer to the center, in a bell curve shape)
The advantage is that it will compute a much smoother version without large (discontinuous) jumps that can take in information with a wider and more consistent radius. It also will be less affected by the differences in size/shape of the areas used.
Think about using Nearest Neighbors on counties: On the east coast they are much smaller than the Midwest, but the number of neighbors is similar and largely affected the geometry of the boundary. Which is more dense? If your kernel radius is 50 miles you'd get a much different answer that described their relative density much more accurately.