To explain the phenomenon, Bungie tapped engineer and "Chief Caching Officer" Mat Noguchi to give both a technical explanation and the "short answer." The short answer is in essence that the game has always used the Xbox 360's hard drive to shorten load times:

The Xbox 360 HDD has a section for games to use called the utility partition. Games can use this section for whatever they want to; Halo 3 uses the utility partition to cache maps as they will load faster off the HDD ... Think of it as an on demand install of Halo 3 to some scratch space on the HDD.

Noguchi's highly technical account is reproduced in full below:

The Xbox 360 HDD has a section for games to use called the utility partition. Games can use this section for whatever they want to; Halo 3 uses the utility partition to cache maps as they will load faster off the HDD than off the DVD. As a side note, the utility partition can be deleted when other games are played. This is why maps can take longer to load when you play another game in between various Halo 3 sessions. (As was the case with Halo 1 and Halo 2.)

So when Halo 3 runs, if a HDD is present, we copy maps from the DVD to the utility partition (on the HDD). Think of it as an on demand install of Halo 3 to some scratch space on the HDD. Halo 3 doesn't actually know where it's running from, so it always assumes it's running from a DVD. This is an unfortunate consequence of new features (namely, install to HDD) being added to the Xbox 360 after Halo 3 shipped. And as a result, it means that even if Halo 3 is already installed to the HDD, it will still copy maps to the utility partition.

So then the real question is why is copying from HDD to HDD slower than copying from DVD to HDD? In the first case, you are reading from one I/O device (HDD) and writing to the same I/O device (HDD). In the second case, you are reading from one I/O device (DVD) and writing to a different I/O device (HDD). In the first case, because we are reading and writing to the same device, the total copy time is the amount of time it takes to read the map plus the time it takes to write the map. Ultimately this is because for the HDD, you read and write through the same mechanism, i.e., the hard drive read/write head, and those reads and writes cannot occur simultaneously through a single mechanism. (If they could, it would be awesome, and I wouldn't have to document any of this. Unfortunately, it's not easy to do for other reasons.) For the second case, because we are reading and writing from different devices, the total copy time is the larger of the total time to read the map and the total time to write the map. This is because we can read from one device (DVD) at the same time we are writing to another device (HDD). There is some overhead because you can't start writing data until you read it, but it is dwarfed by the time to copy a map. (About 128k vs. 500 MB or a ratio of 4000:1.)

There are other factors that contribute to load times, such as preloading data from the map you are about to start so that you don't have to sit through a non-interactive loading screen; the only difference in load times occurs when we copy a map that has not been cached to the utility partition or has been evicted from the utility partition because some other maps were run. Once a map has been cached, the time to load it will be identical to running Halo 3 off the DVD with a HDD.