Evolutionists laugh at the idiocy of this question, but it's related to a real disagreement in taxonomy.
Biodiversity evolves by speciation: species split into more species, and this creates a tree-like pattern of life. Most taxonomists agree that only whole branches of the phylogenetic tree of life should be named as groups; we call them monophyletic groups. Giving names to groups that are not monophyletic can cause misunderstandings. “How come there are still monkeys?” is one of those misunderstandings.
Dick Brummitt of the Royal Botanic Gardens, Kew, argued for taxonomic recognition of partial branches of phylogenetic trees in a short paper, famously titled “Am I a bony fish?”. His argument was that to make bony fish a monophyletic group we’d have to include our own species within it, thus calling ourselves bony fish. That's a fair enough consequence of recognizing only whole branches. I think his concern arises not from the requirement that whole branches should be recognised as named groups, but from the historical baggage we associate with names. In fact, the branch that's descended from the ancestor of all bony fish already has a name, Osteichthyes (which translates as “bony fish”). I think readers were expected to recoil in horror at being called bony fish, much as creationists recoil in horror at being called monkeys. If only the rules of nomenclature would allow us to invent a non-threatening new name for the whole branch (bony vertebrates would do it nicely), I’m sure people wouldn’t feel so uncomfortable.
If we include humans in the ape clade, or apes in the monkey clade, or tetrapods in the fish clade, or hebes in the Veronica clade for that matter, we get this artificial problem; it’s caused by the baggage we associate with the name of the group in popular culture. And in the case of humans, that baggage arises because many think our species is special and separate from other animals. Some even think it was specially created by a god. The monkey clade has radiated from a common ancestor, and we’re one of the products. We are humans, and we’re apes, and we’re monkeys, mammals, tetrapods, bony fish, jawed fish, chordates, deuterostomes, metazoans, all the way down.
But if our concept of apes doesn't include humans, then the taxonomic group "apes" wouldn't be a whole branch of the tree (because the human twig isn't included), and it loses the meaning that evolution brings to biology. I think it's fascinating to understand our species as a highly modified fish, a true story told so well by Neil Shubin; it's a much more useful and interesting idea than a classification that doesn't even hint at our historical relationship.
The tree-like pattern of life is one of the best pieces of evidence we have for evolution. We discover essentially the same tree when we use DNA data as when we use morphology, or chemical attributes, and the order of the origins of different branches matches their order in the fossil record. It makes sense for disciplines of science to be consistent with each other like this; in fact we usually suspect ideas that are inconsistent with other branches of science. So the coming together of taxonomy and evolution is something we should welcome, even if some of the names we use for groups may make (some of) us feel a little uncomfortable.
Brummitt, R.K. 2006. Am I a bony fish? Taxon 55: 268–269.