Naw, Anti-Aliasing has nothing to do with resolution.
It's a bit difficult to explain without pictures, but I'll give it a shot.
Say you have a picture, which is half blue and half red, and the separation is a diagonal line. Pixels are a grid, so instead of a smooth line, you end up with jaggies.
Like so:
bbbrrr
bbbrrr
bbrrrr
bbrrrr
With anti-aliasing, you change the color of pixels near the jaggy spot, by mixing the two colors. With red and blue, this would be purple. This fools your eye into thinking it's a smooth line.
bbbrrr
bbPrrr
bbrrrr
bbrrrr
Think of it as smudging, but only in jaggy spots.