The one variable $\bar \partial$-Poincare lemma is proven in Huybrechts and Forster and so on in essentially the same way: one shows that for a local form $f d \bar z$, with $$g(z) := \frac{1}{2\pi i} \int\_{B\_\varepsilon} \frac{f(w)}{w-z} dw \wedge d\bar w$$
we have $\bar \partial g = fd\bar z$. I can follow the proof on a formal, line by line level, but the manipulations used to obtain the result really don't have any geometric meaning to me, partially because I really don't understand how I should think of $\bar \partial$ except as "some operator with kernel the holomorphic functions, and which we really really want to be locally exact so that we can use sheaf cohomology." My questions here are: is there any intuitive reason the lemma would be true at all, and why should this integral, which suspiciously resembles that of the Cauchy integral formula, be the correct $g$, and is there more geometric intuition for the $\bar \partial$ operator generally than 'it kills holomorphic functions'?