Why do OTA updates require so much space?
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.

Here is a related link that may contain additional information.