I think this is how a physicst would treat spin_c structures: Suppose $(M,g)\cong(\mathbb{R}^n,g_{ij}\mathrm dx^i\mathrm dx^j)$ is a coordinate chart. We can then define $n$ complex $2^{\lceil\frac{n}{2}\rceil}$-"gamma matrices" $(\gamma_{i\alpha}^\beta)_{1\le i\le n,1\le \alpha,\beta\le 2^{\lceil\frac{n}{2}\rceil}}$ satisfying the anticommutation relation $\gamma_i\gamma_j + \gamma_j\gamma_i = 2g_{ij}$, for instance by choosing an orthonormal frame. Next, one defines $2^{\lceil\frac{n}{2}\rceil}\times 2^{\lceil\frac{n}{2}\rceil}$-matrices $(\sigma_{ij})_{1\le i,j\le n}$ by $\sigma_{ij} = \frac{1}{4}[\gamma_i,\gamma_j]$ and checks that
$$
[\sigma_{ij},\sigma_{kl}] = g_{ik}\sigma_{jl} - g_{il}\sigma_{jk} - g_{jk}\sigma_{il} + g_{jl}\sigma_{ik}\ .
$$
Now one obtains a covariant derivative on sections of the spinor bundle: Given $2^{\lceil\frac{n}{2}\rceil}$ functions $s_\alpha$, one sets
$$
(\nabla_i s)_\alpha = \partial_is_\alpha + g^{jk}\Gamma_{ij}^l(\sigma_{kl})_{\alpha}^\beta s_\beta\ .
$$
This is the covariant derivative $\nabla^{(0)}$ defined on page 11 of the referenced paper; it has the property that $[\nabla_i,\gamma_j] = \Gamma_{ij}^k\gamma_k$. This property almost characterizes it uniquely: All covariant derivatives with this property are of the form $\widetilde\nabla_i s = \nabla_i s + iA_i s$, where $iA_i$ is a "$U(1)$ connection one-form" which acts on spinors by scalar multiplication. (This additional factor has to be purely imaginary to be compatible with the inner product on the spinor bundle.)

Now this is fine if we want to treat one coordinate chart, but we should also make our expressions covariant, i.e. give rules how they change when we go to another coordinate chart. It turns out that this can be done, i.e. there is a prescription $s'_\alpha = \Phi_\alpha^\beta s_\beta$ such that
$$
\gamma_i' = \frac{\partial x^i}{\partial (x')^j}\Phi\gamma_j\Phi^{-1}\ ,\hspace{1cm}(*)
$$
coming from the spin representation of $SO(n)$, where the $\Phi$'s can be obtained from the Jacobian $\frac{\partial (x')^i}{\partial x^j}$. However this is actually only a projective representation, and $\Phi$ is only well-defined up to sign - either sign will work, but when we have three different coordinate systems $x,x',x''$, going from $x$ directly to $x''$ might give a different sign than going from $x$ to $x'$ and then to $x''$. This means that we will not be able to do this story on a general manifold: We can make sense of all expressions locally, but the result will depend on the chart. The way out is a "spin structure", a consistent choice of sign for the $\Phi$'s (you can check that this is exactly the same as a trivialization of $w_2$ in the Cech model for $\mathbb Z/2$-cohomology).

However, there is one more thing we can do to make the $s_\alpha$ covariant: For any $U(1)$-valued function $f$, we can define $s'_\alpha = f\Phi_\alpha^\beta s_\beta$ which will still satisfy $(*)$ since scalars commute with anything. Given three coordinate systems $(x^i)_{i = 1,2,3}$, we have matrices $(\Phi_{ij})_{i,j = 1,2,3}$, and we know that $\Phi_{23}\Phi_{12} = (-1)^k\Phi_{13}$. We can get rid of the sign ambiguity if we find functions $(f_{ij})_{i,j=1,2,3}$ such that $(-1)^k = f_{12}f_{23}f^{-1}_{13}$. (In particular, writing $f_{ij} = e^{ig_{ij}}$ for some real-valued function $g_{ij}$, we obtain a $\mathbb Z$-valued fucntion $g_{ij} + g_{jk} - g_{ik} \equiv k\mod 2$, which is a lift of the Cech cocycle defining $w_2$ to a $\mathbb Z$-valued cocycle, as is explained in Equation 2.12 on page 11). However, when we do this the canonical covariant derivative $\nabla_i$ does not transform covariantly anymore; instead, we must replace it by a covariant derivative $\nabla_i + iA_i$ such that $A'_i = A_i + f^{-1}\partial_i f$. In particular, it looks like that $A_i$ give a connection one-form on the $U(1)$-module described by the cocycle $f_{ij}$. This is of course nonsense since the $f_{ij}$ only satisfy the cocycle identity $f_{ik} = f_{ij}f_{jk}$ if $(-1)^k = 1$, i.e. if there is a consistent choice of signs for the $\Phi$'s, i.e. if your manifold is equipped with a spin structure. However, $f_{ij}^2$ *does* satisfy the cocycle identity, and $2A_i$ defines a connection one-form on the corresponding line bundle. (In my comments above, I called this the determinant line bundle $L$).

Also, we see that $s'_\alpha = f^n\Phi_\alpha^\beta s_\beta$ gives a well-defined covariant transformation rule if $n$ is odd, since then the sign ambiguity cancels out. In order for the covariant derivative to transform covariantly, it must be of the form $\nabla_i + iB_i$ with $B'_i = B_i + nf^{-1}\partial_i f$, which is satisfied for $B_i = nA_i$. This covariant derivative is denoted $\nabla^{(n)}$ in the referenced paper.

All in all, a spin_c structure gives a transformation rule for sections of the spinor bundle such that the covariant derivative $\nabla^{(n)}$ transforms covariantly for odd $n$. If you want this transformation property for all $n$, you need a spin structure.