Why does Shamir's Secret Sharing Scheme need a finite field? I read ampersand's question "Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme", where he asked why Shamir's Secret Sharing Scheme uses arithmetic in a finite field of prime order.

The answers to that question explain that a prime field is not necessary for Shamir's scheme, but, rather, that any finite field can be used.  However, they don't really address the other part of the question, namely why do we need a finite field at all?  Couldn't we just use, say, ordinary integer arithmetic instead?

Can someone please explain (in the simplest possible manner) the reason that Shamir's Secret Sharing Scheme uses finite field arithmetic?
The simplest answer is probably to give an example of information leaked when using Shamir's secret sharing over the integers. Assume that we construct a low degree example, defining $q$ to be a linear polynomial with $q(0)=D$ and $q(1)=a_1$. By interpolation you find that: $$q(x)=(a_1-D)x+D.$$ 

Assume that you are given the share corresponding to evaluation at $2$, i.e. $q(2)$. You can see that $q(2)=2a_1-D$. Since $a_1$ and $D$ are integers, given this single share, you learn the parity of $D$.