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.