If you have a heap of sand and remove a single grain of sand, you still have a heap of sand, right?
For any heap of sand worthy of its name, this is true. You can always remove a grain of sand from a heap and still have a heap. But this leads to paradox, for if you continued the process, eventually there would come a point where you no longer had a heap. (unless of course, you started with an infinite heap of sand of a cardinality greater than your ability to handle.)
One can resolve this paradox by noting that the word 'heap' is a little vague, and that we treat this heap as though it were unchanged, when in fact it was changed by a little tiny bit. If we redefined 'heap' to be "a pile of sand that weighs more than a gram", it would obviously no longer be true that you could always remove a grain of sand from a heap and still have a heap. It's not really necessary to redefine words in this way, but at least it reassures us that the paradox really isn't a paradox, and the world isn't about to disappear in a poof of logical contradiction.
In the case of evolution, consider this statement: the offspring of a species are always of that species. Or to put it another way: cats always give birth to kittens. Nothing could be more obvious and true. Yet, taken to the logical extreme, it is a statement that macroevolution is impossible. It's cats, cats, cats, all the way down.
But just as the heap of sand will ultimately no longer be a heap if we keep picking away at it, the lineage of cats turns into something non-cat if we follow it back far enough.
Just as removing a grain of sand does not leave a heap absolutely unchanged, a kitten is not an exact clone of its parents. Its genetic make-up will be slightly different. Looking at cats as a whole, the frequency of genes in one generation will be slightly different from the frequency of genes in the next (unless cats are absolutely perfect, which I confess is a reasonable hypothesis.)
Before, it was the fuzzy word 'heap' that we could wiggle around to dispel the paradox. Here, the fuzzy (so to speak) concept is 'cat'. Or species, really. Just as there's no elegant way to determine when you have a heap that is about to turn into not-a-heap by the subtraction of a single grain, there's no elegant way to describe the boundary between the very first generation of cats and their not-cat ancestors.
We can make an arbitrary human decision that a heap must weigh at least one gram, and palaeontologists do much the same thing when they decide whether a particular fossil is part of species A or species B. Is that skull Homo sapiens or Homo heidelbergensis? But these arbitrary divisions obscure the continuous nature of the transformations: one grain at a time, or one shift in gene-frequency at a time.