The space of derivations on an algebra $A$ should be thought of as the Lie algebra to its "Lie group" of automorphisms, and the Leibniz rule is the infinitesimal version of the fact that an automorphism of an algebra preserves products. In geometric situations $A$ is the space of smooth (or algebraic, or etc.) functions on some space and the infinitesimal version of an automorphism of a space is a vector field. See this blog post for a thorough discussion. (There is also a local story about tangent vectors but it follows along the same lines.)

The more fundamental property is really the chain rule, but note that linearity and the Leibniz rule are equivalent to the chain rule for polynomials, and in an algebraic setting polynomials are the only things available. In a less algebraic setting, e.g. smooth manifolds, it's actually more natural to require the chain rule for all smooth functions; this is closely related to the idea that $C^{\infty}(M)$ is not really an algebra but a smooth algebra.

Let $A$ be an algebra over a field $k$, such as $C^{\infty}(M)$ for $k = \mathbb{R}$. You should think of "points" as meaning $k$-algebra homomorphisms $A \to k$ and "one-parameter families of points" as meaning $k$-algebra homomorphisms $A \to k[t]$ (at least in a more algebraic setting). The intuitive meaning of "tangent vector" is "infinitesimal one-parameter family of points," and algebraically this means a morphism $A \to k[t]/t^2$. The Leibniz rule is equivalent to the statement that this homomorphism preserves products.

The more fundamental property is really the chain rule, but note that linearity and the Leibniz rule are equivalent to the chain rule for polynomials, and in an algebraic setting polynomials are the only things available. In a less algebraic setting, e.g. smooth manifolds, it's actually more natural to require the chain rule for all smooth functions; this is closely related to the idea that $C^{\infty}(M)$ is not really an algebra but a smooth algebra.