Why are finitely generated modules over principal artin local rings direct sums of cyclic modules?
First, all ideals are of the form $(\pi^i)$. Say $(\pi^n)$ is the annihilator of $M$. We can replace $A$ by $A/(\pi^n)$ wlog. Then $M$ has $A$ as submodule, since there's an element that isn't killed by $(\pi^{n-1})$. Now show that $A$ is injective over itself by Baer's criterion. Say $f : \pi^i A \to A$, we need to extend it to $A$. It suffices to show that $f(\pi^i) \in \pi^i A$. But it follows by induction that if an element of $A$ is $\pi^{n-i}$-torsion, it is a multiple of $\pi^{i}$.
Edit: the idea is the same as in Hailong's answer, but you don't need the result of Hungerford.
Let $I$ be the annihilator of $M$, by assumption $I=(\pi^i)$ for some $i$. One can view $M$ as an $R/I$ module and furthermore, embed $0 \to R/I \to M$. But $R/I$ is also principal artin local, so it is a quotient of a DVR by an element (by Hungerford's paper, in particular it is 0-dim Gorenstein. So $R/I$ is an injective module over itself, and the embedding splits. You are done.
The "local Artinian" assumption in OP's question can be removed as the following more general result holds (rings are supposed unital and commutative):
If $M$ is a finitely generated module over a principal ideal ring $R$ then $M$ is the direct sum of finitely many cyclic $R$-modules $R/(a_i)$ with $a_i \in R$, such that $R \neq (a_1) \supset \cdots \supseteq (a_n)$. In addition, the principal ideals $(a_i)$ are unique with respect to this property.
A principal ideal ring (PIR) is a ring whose ideals are principal, equivalently a PIR is a Noetherian Bézout ring.
The aforementioned general result follows from the existence and uniqueness of the invariant factor decomposition of Noetherian elementary divisor rings see [Theorems 9.1 and 9.3, 1]. The fact that a PIR is an elementary divisor ring in the sense of I. Kaplansky is an immediate consequence of [Theorem 12.3 and subsequent remark, 1] and can also be inferred from [Theorem 1, 2]. This fact is stated without proof in [Notes on Chapter I, 3].
By the way, it is rather unfair to allude to Zariski-Samuel's structure theorem (as T. W. Hungerford in [2] and subsequently in the wikipedia article on PIRs) when quoting the result about PIR decomposition in direct sum of domains and Artinian local rings. Indeed, this theorem was originally proved by W. Krull in 1924 and reproved by I. Kaplansky in 1949, cf. [Theorem 12.3, 1] and related comments therein.
[1] "Elementary divisors and modules", I. Kaplansky, 1949 (MR0031470).
[2] "On the structure of principal ideal rings", T. W. Hungerford, 1968 (MR0227159).
[3] "Serre's problem on projective modules", T. Y. Lam, 2006 (MR2235330).