This is possible because iPhone and iPad apps were built on the same silicon used in the new ARM for Mac architecture. This makes them usable across platform, though Apple is unlikely to allow Mac apps over on iOS.
In the part of the WWDC keynote where the feature was briefly discussed (watch it below), Apple exec Craig Federighi said that “most apps will just work with no changes from the developer”, which is a little ambiguous. But if the transition is as seamless as Apple suggests, you should be able to run iOS apps with ease.
Apps appear to present themselves in floating windows, which makes sense for their formatting. It might not be the greatest user experience, but if you want to run something on your desktop that you’d normally pick your phone up for, it’s going to be really useful.
It’s similar to what Google does with Android apps on Chrome OS, and we’re fairly confident that Apple will be able to make the experience decent.