Try drawing a Venn Diagram with intersecting circles to see this intuitively. Label one circle A, the other B. What's the area of the combined figure? It's the area of circle A, plus the area of circle B, minus the overlap area so you don't double-count it. Now try with three circles in kind of a triangle shape. You'll find that you want the area of A plus the area of B plus the area of C, minus the areas where they overlap. But you'll notice that you've overcorrected (you've added in the part where all three circles overlap three times and subtracted it out again three times), so you add it back in again.
I always found diagrams like this very useful when studying basic probability. Now, years later, I remember the pictures better than the actual derivation of the rules!