Why do OTA updates for iOS always require at least 3GB of free space? It's kind of a pain to have to delete apps and then to have to reinstall them again.
The old OS has to run until the new one is in place and ready for the device to restart. Right there, you have 2x the space of a generic OS image.
Then, the OTA updates take more space than iTunes with a computer assisting since your iPhone must have space not only for the compressed image, but also to store uncompressed contents of that download.
Lastly, some amount of temporary space is needed as it actually installs the updated firmware. This space is for upgrade scripts, patching settings, upgrading databases and getting all the old content ready for the new code.
Here's an accounting of the current fall 2013 upgrade with specifics. iOS 6 took up something like 600MB of space, but required something closer to 2.5GB of free space for the OTA update download/installation process, so it should be no surprise that iOS 7 asks for 3 GB space.
When you download OS updates over the air, the update package is usually compressed to save you and the update server bandwidth. Once the download is completed, your device needs to extract the contents of the compressed package in order for the OS to install the update. Thus, it requires that extra space for this step.