The keyboard shortcuts for copy/paste in the iOS simulator can be a bit confusing. This is especially true for Cmd+C and Cmd+V, which most people have deep in their muscle memory. When you invoke these in the iOS simulator, nothing seems to happen, and the titles in the Edit menu aren't particularly clear about what should happen either – they simply say "Copy" and "Paste", like in any other app.
What actually does happens though, is that you're copying between the OS X and the (simulated) iOS clipboard. Cmd+C copies the iOS clipboard over to OS X's, Cmd+V does the opposite (but doesn't paste the result on iOS).
There are a few other useful copy/paste-related shortcuts available, so here's a complete(?) list:
⌘C – Copies the contents of the iOS clipboard to OS X. This works for both text and images.
⌘⇧C – Copies the selected text (or whatever else responds to the
copy:selector in your app) to the iOS clipboard. This one does not appear in the menu.
⌘⌃C – Copies a screenshot of the iOS app (without the simulator's chrome) to the OS X clipboard (Use ⌘S to save it to a file on the Desktop instead).
⌘V – Copies the contents of the OS X clipboard over to iOS, but doesn't paste it. Again, this works for both text and images.
⌘⌃V – Pastes the OS X clipboard in the iOS app, but leaves the clipboard unchanged. This does not work with images.