Linear means called exactly once. Linearity requires proof that the value is definitely used, which is tricky. Affine types are more common in practice, even though people will call them linear.
I wondered the same! It seems to have to do with Girard's idea that linear logic involves 'additive' and 'multiplicative' 'universes' (all words in quotes because I've only skimmed to try to get an idea—even the editors of TCS apparently found the paper unreferee-able). See logical p. 3 (physical p. 4) of http://iml.univ-mrs.fr/~girard/linear.pdf . (How exponentials fit into the picture I don't know.)
It's related to linear logic and affine logic, and both are kinda related to linear algebra I think.
With linear algebra, a function f is linear if, for any a,b, f(a+b)=f(a)+f(b)
Suppose there is a vending machine that sells soda for $1 each (doesn't have a menu, just dispenses the soda, for simplicity). If you put in $1, you get a soda. If you put in $1+$1, you get a soda + a soda.
The amount of soda you get is linear in the amount of money you put in.
This might not be a correct explanation of why it is called linear.
Also I don't really understand how the affine fits in this analogy, other than that "affine" is a somewhat weaker assumption than linear.
I hope someone can give a better answer than I did, because I thought I knew the answer, but when I tried to explain it, I found that I did not really know the answer.
