How do control rods work? I understand the basic idea of nuclear fission: put a bunch of fissionable material together and let the neutrons fly.  An atom gets split, kicking out a few more neutrons, which split other atoms, which release more neutrons, and then you have a chain reaction.
Nuclear reactors also have control rods, which you stick down into the fuel to absorb neutrons, to slow the reaction down, or withdraw to speed the reaction up.  But that's the part that's never made any sense to me.
If neutrons travel from the nucleus of one atom to the nucleus of a nearby atom to split it and perpetuate the chain reaction, doesn't that take place within the fuel itself, on an atomic scale?  How can inserting a macroscopic object nearby affect this process in any way?  By the time the neutron gets out of the fuel to the point where it can strike the control rod and be absorbed by it, isn't it out of the fuel by that point and not going to cause any more chain reactions anyway?  So what do the control rods actually do that affects the rate of the reaction?
 A: This is a very interesting question:

By the time the neutron gets out of the fuel to the point where it can strike the control rod and be absorbed by it, isn't it out of the fuel by that point and not going to cause any more chain reactions anyway?

It reflects a natural intuition about the role of the fuel and the coolant which just isn't how it works. Neutrons in a nuclear reactor are essentially a form of radiation. You probably have a sense that radiation passes through matter. Exactly how far depends on the type, but neutrons can travel particularly far (among those with mass at least) since they're uncharged.
Since neutrons are uncharged, they only interact through nuclear forces, not the electrical environment that permeates electron orbitals. You may know of the analogy of a marble in the middle of a football stadium to represent the relative size of the nucleus versus an atom. This gives us some intuition of how long it will take for some arbitrary neutron to be captured. At atom itself is still very small, but nonetheless, you would have to travel a matter of millimetres to "hit" a nucleus in the most simple sense. In the real world, the effective cross section (area) that a nucleus presents to a travelling neutron is dictated by quantum mechanical effects, so a neutron can partially fly "through" some nuclei and also get "snatched" by some without hitting. But the order of magnitude still puts us in the right neighbourhood. Neutrons travel macroscopic distances because they must hit a scarce nuclei.
Fast neutrons travel much further, and this tends to be on the order of a foot in a reactor which is 14 feet tall. But I don't expect this to convince you yet. The control rods enter the top (in most designs) and this affects the reaction rate all the way to the bottom of the reactor. In the extreme, control rods entering at the top can shut down the entire reactor, but surely not all the fuel is within 1 foot or even a few feet of the rods. That is because diffusion has a role.
You can imagine a fission event at the bottom of the reactor, and a neutron travels upward 1 foot. It gets slowed down and then absorbed there. That absorption creates another fission event, and this one releases a neutron directly upward. Repeat 14 times and you're at the top of the reactor. You could repeat this 14 more times downward and you'll see that fissions at the bottom influences the rate of fissions at the top and vice versa. Obviously the effect is attenuated because of the sheer probability of this type of sequence of events. But as a nuclear fuel assembly, your life is all about giving and receiving neutrons. If your neighbours consistently run a trade deficit beyond your criticality point, the reaction will eventually stop for all of you. 
In practice, for sufficiently long time frames, the total reactor's multiplication factor is always 1.0 as long as its operating. That is accomplished naturally by feedback factors. Most importantly, there is a power feedback factor. Running at higher power places a drag on the multiplication factor. Because of that, when you insert control rods, you will actually reduce the power. Control rods change the shape of the neutron flux and also decrease it overall. Obviously it will depress the flux right next to where it was inserted, but it also reduces the power in all regions by some smaller amount.
A: Short summary
In fact, in typical reactor, neutron needs to travel quite a lot before it initiates next fission, if during these travels it encounters control rod it is "lost" and chain reaction slows down. 
Neutron needs to travel because it needs to lose energy (or in other words slow down), this is because modern reactors are designed in such way that fast neutrons wouldnt be enough to support chain reaction (to know why read the rest!). This is a design decision -- you could have a reactor working on fast neutrons -- it just wouldn't be controlable by control rods! 
To answer most crucial OP question: 
Q: If neutrons travel from the nucleus of one atom to the nucleus of a nearby atom to split it and perpetuate the chain reaction, doesn't that take place within the fuel itself, on an atomic scale?A: Not really due to the fact that you need to slow down the neutrons, neutrons travel macroscopic paths, and during these travels might be lost to control rod. 
Introduction
In typical modern reactors (experimental ones might be different) you do fission by thermal neutrons (thermal means that these neutrons are in thermal equilibrium with the reactor --- that is have the same speed distribution as it should have in working temperature of the reactor --- neutrons produced by fission have much greater speeds).  You can have reactor that works on "fast" (non-thermal) neutrons but these are experimental and much harder to control. 
Not every interaction between a neutron and uranium nucleous will result in fission, moreover probability of fission depends on neutron energy. Basically the lower the neutron energy is the more likely fission is. See this chart (from wikipedia): 

Typical modern reactors
Neutrons produced in fission have high kinetic energy, so before neutron initiates a fission it must lose most of of the energy, so it's free path is quite long (mean free path is length of path that average neutron travels before initiating next fission). Because of that it is improbable that neutron will initiate fission just after it was produced, because it will still have to much energy. Control rods have plenty of ocasions to catch neutrons.
To slow down the neutrons you'll need them to collide with something (like hydrogen atom, uranium atom and so on). However it turns out that when neutrons collide with heavy atoms they tend not to lose energy, they just change direction (this is just basic mechanics not something nuclear related).
Moderator is a material designed to efficiently slow down the neutrons --- this is a material that has a lot of light atoms (water, graphite, helium). Control rods also displace moderator, so neutrons have lower probability of losing enough energy to initiate fission before they escape the reactor. 
You have chain reaction if each fission produces exactly 1.0000 neutron that initiates next fission. To "turn off" the chain reaction you don't need to change this value to 0.0000, not even to 0.9000. Actually if you change the value to 0.99 you'll stop the reaction very fast (I couldn't find precise numbers so this might be off a bit). This is due to the fact that average time between each consecutive fission in chain is low. This means that control rods don't need to drastically alter probability of next fission, just a little bit is enough, even some neutrons wouldn't exit fuel capsule, you need to absorb only tiny fraction to slow down (or kill chain reaction). 
What control rods are for
Last thing: control rods are designed to control the reactor and keep it in steady state (that is a proper chain reaction). If you need to shut down the reactor (becaue of some emergency) other means are sometimes employed --- but these specific details of these vary. For example, in some reactors you can introduce "poison" that is a material that absorbs neutrons way faster than uranium, but does not fission, and so starves the chain reaction. 
Reactors on fast neutrons
OP asked how does atomic bomb work if there is no moderator there. The answer is you can have both atomic bomb and atomic reactor without moderator. Both can have breeding factor equal or greated than one using fast neutrons. Fission crossection (probability that neutron initiates fission) is lower for fast neutrons but it is enough to both have chain reaction and cascade reaction. 
Modern reactors are designed this way that they wouldn't work on fast neutrons --- that is: geometry is designed in a such way that to obtain breeding factor equal to 1.0000 you need thermal neutrons. 
It us really easy to create atomic bomb --- you just need to amass a lot of material in the same place and it'll blow.
Obtaining chain reaction on fast neutrons is much harder --- mostly because of the reaction speed. Fast neutrons take a lot of time to lose energy, so you have additional time to slow down the reaction. If you use fast neutrons you don't have this additional time so each step of chain reaction is -- in order of magnitudes! -- faster. In this case control rods are not fast enough and wouldn't stop cascade reaction. 
People try to create fast neutron reactors, that use much faster control means. One of the design is accelerator driven systems. In this system breeding factor by design is below 1.0 like 0.995 (once again: these numbers are may be wrong but idea behind them is solid) and additional 0.05 comes from an accelerator (or any other neutron source) that injects neutrons into reactor chamber. It turns out that you can control accelerator intensity fast enough to support chain reaction. Such reactors are "hot" research topic in the field (for example because they can work on "burnt" nuclear fuel). 
A: Assume that you have  a fission of an atom of  $U^{235}$, and that we look at one of the neutrons produced.
Although the neutron itself is sub-atomic, the "size" of the space needed for the fission neutron to slow down through collisions with the moderator atoms, avoid capture by control rods or reactor structure, find another atom of  $U^{235}$, collide with that nucleus, and induce another fission, is measured in centimeters or even meters, rather than nanometers.  If the reactor is too small, the neutrons run the risk of leaving the reactor completely. 
Simply put, the control rods are not "macroscopic" in an atomic world;  they are the same sort of scale as the space covered by the nuclear events they control. 
A: Actually, you forget the important duty of the water between rods.
The neutron produced by a fission reaction will have a very high energy. However, if you want it been able to produce an other fission reaction efficiently in the thermal neutron reactor, you need to "slow it down". And this is why you have water. So the neutron will first go through water, slow itself down, and then, it will produce an other reaction.
So now you can understand how the control rods can work : they will absorb the neutron which are in the water.
A: The next atom that splits in a chain reaction doesn't care where the neutron that split it came from, or what path it took to get there. It could have been emitted from an atom right next door and take the shortest possible path, or it might have been emitted from an atom way over on the other side of the reactor and had really good luck with it's flight path, or it might've been emitted from some random atom in the reactor, flown out of the reactor completely, bounced off some random nucleus in a support girder somewhere else in the building, and then flown back into the reactor again. Or innumerable other possibilities.
Not all of those paths are equally likely, but they all could happen.
Inserting control rods into the reactor replaces a lot of atoms of air (or water, or whatever else happened to be filling the space) with atoms of the control rod material instead within some volume which intersects the probable flight paths of some fraction of neutrons through the reactor. That changes the sorts of obstacles (in the form of specific types of nucleii) that those neutrons might run into, be absorbed by, or bounce off of as they travel. That in turn alters the statistical properties of large groups of neutrons who paths take them through the volume of the control rod (total number, speed distribution / temperature, and direction distribution), which controls which parts of the reactor get exposed to how many neutrons of particular energies, and thus how the chain reaction proceeds on a large scale.
