Look at any film on blacksmithing and you will see that after a couple of strikes often only one - on the piece of iron being worked, the blacksmith rests the hammer on the anvil, usually only momentarily. Why is this?
To rest his arm? He has to work all day.
I’ve noticed this too… I thought it might be 1) hammers can get heavy, so little rests help, and 2) it’s a rhythm thing, like tennis players bouncing a ball before serving or squash players tapping the back wall at the start of each shot.
It also allows small adjustments to grip between strikes and gives you a chance to consider your next strike.
IANABlacksmith, but have dabbled with the craft and know a bunch of professionals. Without checking with them & in my experience, using the heavy hammer with enough accuracy simply demands all these mini-breaks. A handful of strikes is all it takes to fatigue the hammer arm and hand into a slop-inducing state, as you need to hit the piece being worked on just so for desired effect. Obviously, coarser work can be done using longer bursts of hammer strikes.
To check the angle between the face of the hammer and the anvil. It is hard to maintain this for more than a couple of blows. Most often the hammer face needs to be parallel to the anvil surface to avoid making dents in the workpiece.
It might also be done while the work is turned or positioned on the anvil.
Several friends of mine are accomplished blacksmiths (I did a little bit of smithing myself, when I worked with them at the ren faire in the 1990s).
It’s going to vary from smith to smith, as they each have their own styles. As others have noted, part of it is a brief rest. But, mostly, it’s to check the work piece. With each strike, the piece’s shape changes, and it moves around on the face of the anvil. IME, you take that little break to eyeball the piece – have I gotten it into the right shape yet? Is it still at the right angle on the anvil (as Kevbo notes)?
As far as why do it with the hammer on the anvil (as opposed to somewhere else), my suspicion is that it’s mostly for economy of motion. From that position, you don’t need to move the hammer much to get back into striking position. And, placing it on the anvil itself gives your arm a tiny bit of a rest.
I have noticed some smiths who do not do this like one of the guys in our group who instructs at the local community college and demos a lot at meeting. I generally don’t (although I am just a hobbyist). I would be curious to see what Uri Hofi (an Israeli blacksmith who has popularized something called the Hofi hammer technique) does. As to why it’s done, I have heard different answers and if someone wants to claim it’s a superstition of the craft, I would have no trouble believing that.
I hesitate to post this because I can’t back it up:
I faintly remember someone telling me that resting the hammer on the anvil also allowed some of the heat absorbed by the face of the hammer from the workpiece to be transferred to the anvil, thus helping to prevent the face of the hammer from heating up too much.
I have my doubts about this theory because I can’t imagine the hammer absorbing enough heat in such a short time to cause any damge to its face.
I wonder if this is a control issue. I have never blacksmithed but when I do repetitive pounding I sometimes do this. I find that it is easier to control the swing when I bring it back from a rest than when I bring it back from a hard swing.
So maybe it is easier for blacksmiths to perform an even strike starting from the same point every time, rather than try to adjust for any “bounce” the hammer may have had from the previous strike.
And on review, the “resting” place will never have moved. Whereas the blacksmith (and me when I hammer) not only have to adjust for bounces, but adjust for the thing they are hitting not being in the same place, if they don’t move it back to the same place every time.
I wonder too if by resting it against the anvil it mutes the ringing created by the blow, much like a bell ringer in a choir will cup the bell to their shoulder to end the tone.
FWIW, I’ve heard this also.
(I presume that we are discussing the momentary “rest” between hammer strikes that lasts little more than the length of time an actual strike takes. If we are talking about simply setting the hammer down for a rest, consider that those hammers can get heavy after a few hours.)
I can’t imagine a rest of less than a second transfering enough heat to matter.
I asked my FIL about this, one time, while he was making a shoe. He said that it is partly about keeping a rhythm going while he examined the piece being worked. He also noted that some smiths hated the practice and would teach their students to avoid it, figuring that it took extra effort and was distracting. OTOH, after 50+ years of doing it himself, he doubted that it made much difference one way or another and that it was as much nervous habit, (emphasis on habit, not nervous), as anything else.
My first thought was akin to this. Vibrations transferring up a tool handle can make using it more tiring, as well as reducing control.
Professional blacksmith here, kenobi 65 got it right. We’re looking at the piece to check its shape.
Heat in both the hammer and the anvil is a good thing, since a cold hammer and a cold anvil will draw heat out of the work, causing it to cool faster and requre re-heating (more heats) more often. As in any profession, time is money.
A ringing anvil and vibration in the hammer are not big deals, and can be dealt with in a number of ways if they are bothersome to the smith. Resting a hammer on the anvil wouldn’t dampen the ring much anyway.
Short answer to the OP: Where else would you have us rest it? 
I love the Dope!