There are many words and sentences in mathematics that I basically completely don't understand, including the words "Koszul" and "derived". But rather than ask for a complete description of such words, I will ask about a particular example, and hope that an MOer can spell out concretely what's going on. (Of course, links to good write-ups of Koszul, etc., are always welcome.)

There is a relatively easy theorem, probably due to Chevalley and Eilenberg, with many generalizations due to Koszul:

Let $L$ be a (finite-dimensional) vector space (in characteristic $0$); let $[1] L^*$ denote the dual vector space, "shifted" to a graded vector space supported in degree $1$; and let $([1]L^*)^{\vee \bullet}$ denote the free graded-commutative algebra generated by $[1]L^*$ (I impose the Koszul rule for signs, so that as an algebra $([1]L^*)^{\vee \bullet} = (L^*)^{\wedge\bullet}$ is classically an alternating algebra). Then to give a Lie algebra structure to $L$ is the same as giving to $([1]L^*)^{\vee \bullet}$ a square-zero degree-$1$ derivation.

The construction is (contravariantly) functorial and full and faithful, and so embeds the category of Lie algebras fully-faithfully into the (opposite to the) category of commutative dgas.

Here are two examples. I will call my characteristic-$0$ field $\mathbb R$. The one-dimensional Lie algebra corresponds to the dga $\mathbb R \overset 0 \to \mathbb R \xi$, where $\xi$ is the coordinate function on the one-dimensional vector space, and the algebra is graded-commutative, so $\xi^2 = 0$. The two-dimensional non-commutative Lie algebra corresponds to the dga $\mathbb R \overset 0 \to (\mathbb R \xi \oplus \mathbb R \upsilon) \overset{\xi \mapsto 0, \, \upsilon \mapsto \xi\upsilon}\longrightarrow \mathbb R\xi\upsilon$.

Now, once we're in the land of dgas, it makes sense to talk about their (co?)homology. In the above examples, the cohomology of the one- and two-dimensional Lie algebras are isomorphic as algebras; the isomorphism on cohomology is given in one direction by the "abelianization" map from the two-dimensional Lie algebra to the one-dimensional Lie algebra.

Question:How should I interpret "geometrically" the fact that the cohomologies agree?

An idea that I've heard is that somehow dgas-up-to-? should correspond to some notion of "stack". Now, it is certainly not the case that "point mod one-dimensional" and "point mod two-dimensional" present the same stack. So that's not quite what's going on. Perhaps the problem is that these algebras are not the same as ${\rm A}_\infty$ algebras; we should probably add the word "${\rm A}_\infty$" to the list of words I don't really know. If this is the answer, I hope that some MOer will spell it out.

Another idea that I've heard is that the passage "Lie algebra to dga to cohomology" remembers exactly the "derived category of representations of the Lie algebra". Again, I don't really know what that is, but that's OK. Somehow, a representation of a Lie algebra should be a "quasicoherent sheaf on point mod the Lie algebra", and I know that people like "derived categories of quasicoherent sheaves". So should I understand Koszul duality as remembering not all the data of some "stack-like object" but just some "derived data"? If so, again I hope that some MOer will spell it out pedantically in this example.