If your arms are resting, then your fingers and wrists are doing the maximum amount of reaching as you type. If you use a wrist rest you are encouraging your fingers/wrist to reach up (bend in your wrist) instead of neutral or reaching down (more natural position).
Constant change in position, dynamic motion, strong & brief muscular exertion of whole hand and arm, even including the shoulder vs constant static position w/ small repetitive motions, where the fingers really are doing almost all the work.
You should try to avoid continuous static load on your muscles, especially the smaller ones. So you should find a typing position where that doesn't happen. You also want to use your muscles in the strong and comfortable part of their range of motion, which depends on the entire chain of joints, because tendons have to stretch past several joints to get to whatever bone they attach to – so for fluent finger motions, you want to keep wrists and hands in as neutral a position as you can.
If your wrists are not straight while typing a lot, that's really bad. I constantly see people typing with their wrists either significantly flexed or significantly extended; doing that a lot is a fast road to RSI, and even doing it a little is pretty unpleasant and inadvisable.
If you are going to type a whole lot at a stretch (say, as a programmer or writer), you want your arms to be mostly passively supported from the shoulder. Having your arm bent at the elbow doesn't cause much strain, as long as the upper arm is hanging loosely down with your shoulder relaxed – so bring the keyboard relatively close to your torso. Resting your wrists, palms, or forearms on some surface and then typing generally causes more strain than having your wrists and palms "floating" above the keyboard while actively typing. You can rest the fingertips lightly on the key tops if you want. You can rest your palms on a palmrest or arms on an armrest (or table, or lap, or whatever) while you are taking a break from typing. It's generally a good idea to take regular breaks.
There doesn't appear to be good evidence that this is better for RSI. I found one study that shows greater shoulder and back strain when doing the hover hands approach. It might make more sense for piano since you need more mobility up and down the keyboard, but for typing your hands don't need to move nearly as much, so resting should be just fine. The do suggest some sort of support under your wrists, though.
Huh, seriously? Have you ever worked in an office? Perhaps your mental picture of what op is describing might be misaligned? I just always assumed it was a rarer/ more disciplined style some people had
If your arms are resting, then your fingers and wrists are doing the maximum amount of reaching as you type. If you use a wrist rest you are encouraging your fingers/wrist to reach up (bend in your wrist) instead of neutral or reaching down (more natural position).