A utility function is additively separable if it can be written as: $U(x,y) = u(x) + v(y)$
Examples $U(x,y) =ax + by$ is additively separable by inspection. $U(x,y) = ax + bx2 + cy$ is also. $U(x,y) = xa yb$ is additively separable, because you can write it as $U(x,y) = a log(x) + b log(y) = au(x) + bu(y)$
$U(x,y) = \frac{xy}{x+y}$ is not additively separable because there's no way to transform it into an independent sub-function of $x, y$.
Even if you take logs, you're down to $U = log(x) + log(y) - log(x+y)$ - notice that the third term can't be 'split'. Generally, mixing addition, multiplication and exponentiation will destroy additive separability.
And so we can see, The actual definition of additive separability is:
A function $f(x_1,...,x_n)$ is AS if it can be rewritten as $f(x_1,...,x_n)=f_1(x_1)+...+f_n(x_n)$
The assumption is usually one of mathematical convenience. For example, when utility is additively separable in $x,y$, then the marginal utility of $x$ does not depend on the level of $y$, and vice-versa. And so anything dealing requiring us to use partial derivatives is made much easier. IS it a reasonable assumption? Sometimes. For example, if your utility depends on apples and horses, we can probably assume additive separability. If Instead your utility depends on two closely related things, it might be a bad assumption.