- More mathematical way
The variance of two variables, X and Y, can explained by separating it into four sections: how X varies independently, how Y varies independently, how X varies due to changes in Y, and how Y varies due to changes in X. Essentially, all changes in X and Y have to be caused by one of these four things.
That gives us:
$Var(X) + Var(Y) + Cov(X,Y) + Cov(Y,X)$
which can be simplified, because $Cov(X,Y) = Cov(Y,X)$ into:
$Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y)$
- Not mathematical way
This is a vast oversimplification and is not mathematically correct, but it helped me with remembering.
Consider the much simpler algebraic equation:
Also consider that: