In addition to @AliceD's excellent answer, I would like to add that a simple mechanistic relationship between body size and "snappiness" may explain the observed pattern.
Basics of biophysics
Difference in snappiness may result from a difference in the ability to accelerate your movement.
An increase in body size over one dimension (=body length) $x$ scales with an increase in volume and mass of the order of $x^3$. Muscle strength is dependent on the cross-sectional area of the muscle and therefore by a linear increase of $x$, strength increases by $x^2$.
Digging back to high school physics lectures, $F=Ma \left(\text{or } a=\frac{F}{M}\right)$ , where $F$ is a force, $M$ a mass and $a$ an acceleration. With an increase $x$ of the individual size (over one dimension, the length of the individual if you prefer), the force increases by $x^2$ and the mass increases by $x^3$, therefore the acceleration gets multiplied by $\frac{x^2}{x^3} = \frac{1}{x}$. Of course, nature is not all that easy, but if we assume it is, then the relation between the acceleration of movements and body size (=body length) should look like this:

An animal that is 4 times as big should make accelerate its arms and legs 4 times less.
Contrasting to @AliceD's answer
@AliceD listed plausible hypotheses for why there might have a selection for snappy movement in small animals but simpler explanation based on the physical properties of bodies might be enough to explain the observed pattern. It would be interesting to know exactly what the relationship between body size (=body length) and "snappiness" look like in nature to further this discussion.
Fun Facts
The mantis shrimp (see below picture) is able to accelerate its claws up to 10,400g (reference). More information about g (unit of acceleration) here. In short, a typical human can support up to 5g. 10,400g is absolutely insane! To offer some comparison: a baseball gets an acceleration that is only about a third of that of the mantis shrimp's claws. Mantis shrimps use their claws to break open clams. Here is a video. According to wikipedia, due to the friction of the claws against the water, small bubbles are formed and the temperature at the edge of those bubbles can go up to several thousands of °C.

Apparently, a jellyfish stinger can go up to 5,400,000g (reference) which appears to be about 30 times more than a bullet. You will find a list of examples of things that undergo various levels of acceleration on this wikipedia list.