Motivation for Koszul complex Koszul complex is important for homological theory of commutative rings.
However, it's hard to guess where it came from.
What was the motivation for Koszul complex?
 A: In this answer I would rather focus on why is the Koszul complex so widely used. In abstract terms, the Koszul complex arises as the easiest way to combine an algebra with a coalgebra in presence of quadratic data. You can find the modern generalization of the Koszul duality described in Aaron's comment by reading Loday, Valette Algebraic Operads (mostly chapters 2-3).
To my knowledge the Koszul complex is extremely useful because you can use it even with certain $A_\infty$-structures arising from deformation quantization of Poisson structures and you relate it to the other "most used resolution in homological algebra", i.e. the bar resolution. 
For a quick review of this fact, please check my answer in Homotopy equivalent chain complexes
As you can see it is a flexibe object which has the property of being extremely "explicit". This helped alot its diffusion in the mathematical literature.
A: I don't know the historical origins, but it is not so hard to make up a story:
Consider the basic example
$$0 \to k[x] \to k[x] \to k \to 0,$$
where the middle arrow is mult. by $x$.  This is a resolution 
of $k = k[x]/(x)$ as a $k[x]$-module.
Now suppose you want to generalize this to obtain a resolution of $k = k[x_1,...,x_n]/(x_1,...,x_n)$ as a $k[x_1,...,x_n]$-module.  It is not hard to see that you need "one copy" of the above sequence for each variable; tensoring these all together over $k$ gives you the usual Koszul resolution of $k$ over $k[x_1,...,x_n]$.
It is not hard to pass now to the more general context of elements $a_1,\ldots,a_n$ in a ring $A$, and to imagine the the Koszul complex of $a_1,\ldots,a_n$ will related to the module $A/(a_1,\ldots,a_n)$.  
