Why do small animals appear to move faster than larger ones? I am keen to understand why smaller creatures move relatively faster than larger ones. Not only do they move faster, but their metabolism runs at a faster rate, they seem to process information faster (try swatting a fly!) - their entire lives seem to burn more rapidly living shorter lifespans. I’ve read about Kleibers scaling law, but this explanation alone does not seem to offer a fully satisfactory answer.
 A: OK, I think that I can interpret this question in this way
"Why do smaller animals seem more active than larger animals (like a rat seeming more active than a dog)?"
Yes, the point that you have noted is quite correct, their metabolic rates are quite fast- (I disagree to the point that this is a question for Biology SE).
 Assume that you have a small mouse and a larger animal like an elephant. Now the ratio of Surface Area to the body mass of the mouse will be much larger than that of the elephant, which is fairly obvious 
Mouse-$0.875 cm^2g^{-1}$
Elephant (Asian)-$0.0425 cm^2g^{-1}$ 
(These are estimates I made from data off a few papers, the original values may deviate, but not appreciably)
Naturally, a larger surface area leads to greater loss of heat to the surrounding. 
$$\frac{\Delta Q}{\Delta T} \propto A(T_b-T_s)$$
You can see from the above equation, that the heat lost depends on the difference in temperature (known as Newton's law of cooling in physics), and on the surface area of the organism. We take the ratio rather than the total surface area here because a small change in heat will produce a larger change in the body temperature of smaller organisms.
So, in order to maintain their body temperature constant, there is a need for active and constant metabolism. And constant metabolism naturally implies that they must be more active (which means that that they seem to move relatively faster) and need a continuous supply of food to compensate for this process. 
But this applies mainly for regulating organisms (i.e. organisms that maintain a constant internal temperature) such as mammals and small birds. You might find a small mammal in the tropics (where $T_b-Ts$ is less) but not in Antarctica, (where $T_b-Ts$ is large).  
So, I feel that the point about swatting a fly (or processing information faster, for that matter) is purely an evolutionary incident as opposed to metabolic rates. Lower organisms have a reflexive body i.e. they rely more on reflex signals and the apparent inability to hit a fly is a reflex action- in any case, it is too much to expect information processing in a primitive brain.
A: As the Wikipedia page on Kleiber's law explains, metabolism is in animals a volume property. However, animals interact with their environment through surface phenomenon. This interaction can be in the form of heat loss/gain or in the form of resources acquisition/waste elimination. Many physiological processes will depend on diffusion and/or fluid dynamics. Long pipes and tissues far from external surfaces will be limited by these physical processes. Note that "external" surfaces can be increased through geometry. For example, the inside of the lungs contains air and can be considered as an external surface. 
Now, this does not mean that small animals have to process resources faster. They physically (in the sens of how physics allows it given the right structures) can, but whether they do or not is a matter of what their environment is. There are relatively small animals that have a very slow metabolism. See for example olms https://en.wikipedia.org/wiki/Olm. These small (1 foot long) salamander-type organisms take 14 years to reach sexual maturity. These critters can also survive 10 years without food if needed. 
However, Kleiber was mainly concerned with mammals. Being warm blooded, these animals have to expend energy to keep their temperature within a predetermined range, or else they die. Smaller animals have an advantage in this respect (ie having a large metabolic rate if desired) as they can spend more energy per unit volume than larger ones. If more active animals can reproduce more, evolution will select for larger metabolic rates in small mammals than in large ones. 
To get closer to your fly swatting example, a big advantage of smaller animals is that there is a smaller distance between their cells. A lot of information processing is performed by nerves in complex animals and nerve conduction speed is of the order of meters per seconds. The distance between the fly's eyes and its wings is much smaller than the distance between your eyes and your arm. Even if there were no processing delays (ie, signals move uninterrupted between sensor (eyes) and actuator (muscles)), the fly will be able to adjust much quicker than you to sudden unexpected stimuli. That is, the fly can process very quickly that an object is coming towards it, while you take a long time to realize that the fly moved. Also, simple animals, such as flies, have very simple nervous systems and actions a reflex-based, as opposed to complex processes in many human tasks. Such complex processes require a large number of neurons, often in different parts of the brains. By the time you realize the fly moved, it is already gone. 
The last piece of the puzzle here is that smaller animals have of course a smaller mass. If they can release a lot of energy in a small amount of time (relative to their mass), they can theoretically move very fast. Releasing energy will require moving some body parts. However, you don't want to break those body parts while moving. structural strength will scale with cross-sectional area, while forces, which depend on inertia of the animal and/or body parts, will scale with volume. Here again, small animals are at a major advantage as they will be able to grow structures that can transmit the forces required to move their body quicker. 
Now, let's get all this in perspective. I will make an analogy between swatting a fly and hitting at baseball. This is not perfect, but there is a lot of data on baseball, less so on fly swatting. Also, this is only an order of magnitude approximation. Let's say you swing your swatter at 10 m/s. This is a bit less than throw speed for a pro pitcher, but you are not using all you can to accelerate your swatter either. Let's say also that the fly sees that it is getting into problems when you are 1 meter from it. Then, the fly swatter would hit the fly in 0.1 secs. Reaction time of pro baseball hitters is about 200 ms. That is, they take .2 s to adjust their swing to the incoming ball. You are in trouble. If the fly moves outside of the area of your swatter, you have no chance of hitting the fly. 
Now, let's assume the fly only has to move 10 cm (0.1 m). A fly has a reaction time of about 20ms (see for example https://medium.com/the-physics-arxiv-blog/the-creature-with-the-quickest-reactions-might-surprise-you-da3752e279). This is commensurate with 1 cm distance between eyes and wings and a speed of information propagation of meters per seconds (all orders of magnitude). To accelerate at a constant rate, one can use kinematics equations to see that if the fly can accelerate at a bit more than 2 g for .08 s (.1 s - reaction time), it will be able to escape your swatter. You have probably observed flies circling around you. Let's say a fly circles around you in 2 seconds in a 2m radius circle (seems reasonable). This gives a centrifugal acceleration of about 2g. Since flies seem to be able to pretty easily accelerate at 2 g, the fly has the physical ability to escape (as we already observed).
OK, one last observation. Since reaction time is 0.02 s in flies, and the fly has to run away at a limited speed, the way to be certain to hit the fly is to hit it from a small distance. For an immobile fly, here is the trick. Flies have very simple nervous systems, and basically only have an escape reaction to fast movements. I can catch flies with my hand if I approach them slow enough (you must get close to them really slow, over 10 seconds or more). Think about optical flow in the fly's eyes. The closer you are, the slower you have to go. Now, if you can get your swatter close enough and accelerate it very fast, the fly will not have time to go out of the impact zone before it gets squashed.  
[EDIT]
As Charles Francis mentioned: muscles also produce forces proportional to cross-section, while mass goes as to volume cubed. Therefore, smaller animal can produce larger accelerations than larger ones. This is also in favor of flies escaping.
A: There are essentially two reasons. First, as they are smaller we are likely to look at them from nearer, creating the illusion of greater speed which results from greater angular speed subtended at the eye (just as when travelling by train or car, near objects cross our field of vision more quickly).
Second, they are lighter, body mass going with the cube of linear size. Muscular force goes with the cross section area, that is the square of linear size. Consequently they do in fact accelerate much more rapidly than larger animals, and greater acceleration can also produce an illusion of greater speed. 
