To expand on Mariano's comment, let $A$ be, say, an $\mathbb{R}$-algebra and $\phi : \mathbb{R} \to \text{Aut}(A)$ a one-parameter group of automorphisms of it. Suppose that we can make sense of the derivative of $\phi$. Then letting $D = d \phi_0(1) \in \text{End}(A)$ we find that differentiating the condition that $\phi$ is a family of automorphisms gives that $D$ satisfies the Leibniz identity. In other words, the Leibniz identity is the infinitesimal analogue of preserving multiplication. (For example, the standard derivative of a one-variable function is the derivation associated to the one-parameter group of translations.) 

In fact $A$ need not be associative for this argument to go through. This makes the above a nice way to think about, for example, the Jacobi identity for a Lie bracket: it says precisely that $[x, -]$ is a derivation, and this is simply because it is the infinitesimal version of the adjoint action.