Why are imports subtracted from GDP? GDP is a measure of a country's production.

$$GDP = C + I + G + X_n$$

$C$ = Consumer Consumption
$I$ = Gross Investment
$G$ = Government Expenditures
$X_n$ = Exports - Imports

Exports are what we produce and make a profit from by selling to buyers outside our country.  Imports are not produced by our country, so it shouldn't be included in the GDP, so it makes sense to exclude it from the calculation; ie. there should be no "- imports" in the calculation.

However, the calculation subtracts imports from the GDP.  Imports somehow take away from what we produced?  That seems to say, calculate how much I have produced, X, and then don't count some of it because I imported Y.  Which doesn't make sense given that importing doesn't remove goods and services that have already been produced!

For example, let's say I can take apples and make pies... I produced value in the form of the "pie" quality.  Importing the "pie" quality and tacking it on an apple creates an apple pie.  However, I didn't make that "pie" so I can understand how it doesn't get included in the GDP: the value of the "pie" quality was not produced in this country, so it isn't included in the GDP.  

Let's also say my GDP = "made a cat meow" + "turned a tree into a fountain".  Somehow, by importing the "pie" quality, I am to ignore some of the value of making a cat meow?
When we import something, we consume it. So when calculating consumption we are bound to count import as a positive component of GDP. Since it is not (we did not produce imports domestically), we subtract it to make it neutral.