On the one hand they are integral components of a character’s interaction with the game mechanics.
But on the other hand they are also carriers of condensed descriptive information about a character.
They are a shorthand for describing a character’s personality, background, and aptitudes.
Not mere numbers or labels, but talking ones.
They are codes.
Obviously, the relative value of these codes depends on the underlying system (besides the user’s skills in assigning and interpreting them).
The host of White Wolf Storyteller/Storytelling games offers a good example of a rules set enabling this coding feature of stats.
With each dot in each possible value underlaid with a description of what this is meant to represent outside a rules context, what level of training and expertise this is associated with, and in many cases even including examples of who might be expected to possess such.
All together, this means a single line, consisting of a one or two words and a set of dots, conveys a whole host of information, that can run from physical condition and age, to professional status, education, social circles and more. The combination of statistics allowing to construct an ever more complex picture, without ever leaving the confines of the character sheet.
Another excellent example would be the so called character profile of Blue Planet (both v2 and the newly arrived Revised).
Consisting of three keywords, the profile is used to describe the outlook on and goals in life of a character, in an ultra-compact format.
Note how this differs from freely choosing and assigning three words to describe the same. By drawing the keywords from a closed list, it means that each term is indeed just a code for a longer more detailed description – one that any other person also familiar with the “coding dictionary” used (or having access to it in written form) probably has a far better understanding of the character than in case of the freely chosen words, even if selected to be largely self explanatory.
A high volume of meaningful information delivered in a compact format, fully accessible based on the use of a common dictionary – I don’t need a lengthy background story, I need thought out character sheet.