Say we have a rank $r$ smooth vector bundle $E\to X$ and a smooth section $s:X\to E$ of it. Shortly after the 30 min mark in this video Joe Harris defines the $r$th Chern class of this bundle to be the cohomology class $c_r(E)=[x:s(x)=0]$ of points where the section vanishes.
I believe this is the Poincaré dual to the homology class defined by the vanishing locus of the section, and I'd like to know how to explicitly build this homology class, and seeing why it has degree $2r$.
You should be assuming the section is transverse to the zero section. In that case, transversality says that the intersection is of $4r = \text{codim}_{TE} X \cap s(X) = \text{codim}_{TE} X + \text{codim}_{TE} s(X) = 2r+2r$. (Note that $E$ is a complex vector bundle, so the zero section has codimension $2r$, not $r$!) Then $\dim X \cap s(X) = n+2r-4r = n-2r$. This is an oriented submanifold; triangulating it with simplices whose orientations agree with the manifold. The sum of these is a chain in $X$, giving a homology class of degree $n-2r$. Poincare duality sends it to a cohomology class of degree $2r$, as requested.
As for why this is well-defined: Given two different transverse sections $s_i$, there is a transverse section $s_t: X \times I \to E$. So the resulting submanifolds $X \cap s_i(X)$ are cobordant via the oriented cobordism $\cup_t X \cap s_t(X)$. Triangulating this cobordism shows that the two homology classes the submanifolds $s_i(X)$ represent are homologous.