Why does using foil on polynomial expressions match so closely to integer multiplication?

 FOIL is a mnemonic rule for multiplying binomial (that is, two-term) algebraic expressions. 
FOIL abbreviates the sequence "First, Outside, Inside, Last"; it's a way of remembering that the product is the sum of the products of those four combinations of terms. 

For instance, if we multiply the two expressions 
(x + 1) (x + 2) 
then the result is the sum of these four products: 
x times x (the First terms of each expression) 
x times 2 (the Outside pair of terms) 
1 times x (the Inside pair of terms) 
1 times 2 (the Last terms of each expression) 
and so 
(x + 1) (x + 2) = x^2 + 2x + 1x + 2 = x^2 + 3x + 2 
[where the ^ is the usual way we indicate exponents here in Answers, because they're hard to represent in an online text environment]. 

Now, compare this to multiplying a pair of two-digit integers: 
37 × 43 
= (30 × 40) + (30 × 3) + (7 × 40) + (7 × 3) 
= 1200 + 90 + 280 + 21 
= 1591 

The reason the two processes resemble each other is that multiplication is multiplication; the difference in the ways we represent the factors doesn't make it a fundamentally different operation.