How can we see an atom now? What was the scale of this equipment? I've just seen this on the news - Single Trapped Atom Captures Science Photography Competition's top prize.

Credit: David Nadlinger via EPSRC
I am not a Physics major but I believe I do know the basics. I have always believed that we can't really see single atoms with naked eye. What allows that picture to make us see a single atom?
If that Single atom is being held there by a field, why are the atoms of that very field not visible?
 A: You can't see a lightbulb hundreds of meters away from yourself, even in broad daylight. However, at night, when turned on, you can.
We don't see the single atom, we the light emitted by the atom.
A: To be fair, this is actually explained in your link. To put it simply,

If you illuminate it with the right light, it starts shining so bright that a good camera can detect it.
To make it work, the atom has to be as motionless as possible. This is achieved by "freezing it" and using magnets to hold it still.

Close-up for completeness:

A: While the physics has already been covered in other answers, let me give you an idea about how to explain the difference between detection and resolution to a 4-year old:
Try an analogy. Something you can't resolve individually but see pretty easily.
The fist thing that comes to mind is lights at a distance. A bunch of LEDs at a distance might do it, your computer/TV screen, one of those big screens you can find on buildings, the lit (or dark) windows of a far away house, letters on a piece of paper and probably a lot of things I can't think of now.
The principle stays the same: Choose the right lighting conditions and the right distance and it is easy to see, if a single "pixel" is lit or not. But can you distinguish between one pixel or two? Can you count the pixels if all are lit (a computer screen is probably perfect for this one)? Can you tell where one pixel ends and where another begins?
Ok, the analogy does not explain the resolution limits, but I think with a 4-year old you can get quite a good feeling for the difference between detection and resolution, and for "if I look closer, I see more details - but maybe I can not look close enough without a lot of effort".
A: The questions of whether you can detect light emitted from an (isolated) atom and whether you can resolve an atom from its neighbours are completely independent.
The spacing between different atoms in a regular material remains impossible to resolve using visible light, whose wavelength is several thousand times larger. You can "see" individual atoms by using other microscopy techniques (so see e.g. this short film for a nice example), but those are using rather elaborate instrumentation and post-processing, and they do not reflect what is visible to the naked human eye.
The picture you're quoting, however, does not image one atom out of many in a material. Instead, it really is a single isolated atom, held in a vacuum by a set of electric "tweezers" called an ion trap (itself produced by the metal electrodes that surround the atom, which will be a couple of centimetres across), and which is emitting light via fluorescence (i.e. it is being excited by a laser and re-emitting that light). The size of the atom as it appears in the picture has nothing to do with its actual size: as far as the camera is concerned, the atom is a point source, and the nonzero spread in the image is caused by the finite resolution of the camera.
Thus, assuming that the trapped atom is bright enough, it could in principle be seen with the naked eye, in which case it would look much like a star on a clear, still night (which are also point sources as far as our eyes are concerned, though their appearance then gets changed by twinkling). Whether the experimental configurations in actual use are enough to produce atoms that are bright enough to see with the naked eye is a good question; my understanding is that this isn't quite possible, but that with a completely dark background it isn't that far out of reach. 
That does mean that a human wouldn't be able to see both the atom itself and the trap electrodes simultaneously, since you require a completely dark background to begin to have a chance at seeing the atom. As for the camera, the author has clarified in a comment that it's a single thirty-second exposure, with the electrodes illuminated by a camera flash halfway through the exposure.

Finally, to address your expanded question, 

If that single atom is being held there by a field, why are the atoms of that very field not visible?

the answer is that the field that is holding it up is not made of atoms at all. The atom in the picture is being held in place by electrostatic forces, which are the same forces that you use to pull up bits of paper with a balloon that you've rubbed against your hair. Electrostatic forces, like magnetic forces and gravity, are said to form a field, but it's a force field that's all force and no atoms. The effect here is analogous to magnetic levitation, except that you use electric fields (carefully engineered ones, produced by the metal electrodes that surround the atom in the picture) instead of magnets.
A: When I just explained to my 8yo what atoms are (it turned out she already knew, they seem to be taught that little nugget of information at school these days before that age, anyways), I just took the good old greek approach - imagine to take a piece of cake, split it in two etc. etc. I found that an 8yo is able to understand the concept that at some point you cannot divide "stuff" anymore. I stopped there because a) she does not yet need to know about protons, electrons or even quarks and b) the image of the old philosophers is still true. Splitting macro-level matter (atoms) this way until you hit the atomar (or molecular) level is fundamentally different from then going on and further dividing atoms into their constituents.
For a 4yo, you have to simplify this a lot. Just use scale. Show them a little grain of sand which you can just barely see with the eye. Then put that grain of sand on a table and walk away with your child. Point out how she now cannot see it anymore. This should give her an idea that it matters how close you are to the object.
If you have a magnifying glass in your house, you can demonstrate how big the grain of sand looks if viewed through that.
For the rest, it's a simple analogy: "Now, atoms are just like this, but much much much smaller. And in the past, when I told you we cannot see them, there were no magnifying glasses around which were good enough. But recently, they invented some that can!" Or if that sounds like a lie (I don't know whether there were any breakthroughs in this area in the last 1-2 years since you told your child...), say something like "Sure, those guys have magnifying glasses that are as large as a room, they can see atoms, but we here cannot."
You and me both know that this is a gross oversimplification, but we are talking about the world view of a 4 year old. She does not yet need a complete understanding of quantum effects, photons interacting with atoms and such stuff. The above will give her the concepts, is not totally wrong, and can be approached in more detail much later.
Oh, and obviously there are plenty of humans out there who are adolescent or even adult and have not the slightest chance at understanding the "real" explanations anymore than a 4 year old would. For those, similar explanations can just work fine as well. 
A: In typical human-level experience, atoms aren't alone. Even the atmosphere has huge amounts of atoms in a tiny volume. When you shine light bright enough to see on a volume of air (or a solid object), the light is reflected from large amounts of atoms at the same time. Human eyes cannot distinguish between light from individual atoms, they blend together. When you look at a TV from up close, you can see individual pixels (or even sub-pixels) - when you look from further away, you still see all the pixels, but you can't distinguish them from one another easily.
Now imagine that of the millions of pixels on the TV screen, all are black except for one. Even though the individual pixels are "too small to see" (distinguish from other pixels nearby) normally, one pixel stands out clearly against the black background. And that's basically what these guys have done with atoms - keep a single atom far away from other atoms, and shine a light. The atom is still far "too small to see", but it shows as a blurry spot (much larger than the atom itself) since there's little noise around it. And just like with the pixels on the TV, if you put two atoms close together, you wouldn't see any difference - it would still look like a single atom. In fact, you'd need very large amounts of atoms together to make it obvious that you're no longer looking at the "visible single atom".
In other words, "too small to see" doesn't literally mean too small to see. When you look at light reflected from any object, you're "seeing" atoms, molecules and even individual electrons (as much as you can talk about the identity of an electron anyway - think more about "an electron in a particular orbital", not "electron named Bob"; electrons don't really have identities). The literal meaning of the statement is "too small and too close together to be distinguished from one another". If you look from an airplane at night, you can clearly see the illuminated cities without being able to resolve individual street-lights - even though all that illumination comes from such light sources.
A: Get him a magnifying glass, and maybe even a kid's microscope.  He still won't be able to see atoms, but he'll get the idea that there are things which are too small to see by eye, but which you can still see with the right amount of magnification.  Then all you need to say is that these guys have "really good, really expensive microscopes".
