If $x$ is a root of $p(x;A)$, then $Ax-I$ is not invertible, and therefore has a nonzero null space. So there exists a nonzero vector $\vec{v}$ such that $(Ax-I)\vec{v}=\vec{0}$
So $Ax\vec{v}=\vec{v}$. Note that $x$ cannot be zero for two reasons - we have specified $\vec{v}\neq\vec{0}$ and $\det(A\cdot0-I)\neq0$. We then have that $A\vec{v}=x^{-1}\vec{v}$, and $x^{-1}$ is an eigenvalue for $A$.
So the roots of $p(x;A)$ are inverses of the nonzero eigenvalues of $A$. (If $A$ has zero as an eigenvalue, then $p(x;A)$ has degree less than $n$.)
A more complete picture added later:
If the Jordan canonical form of $A$ (over $\mathbb{C}$) is given by $A=PJP^{-1}$, with $J$ a composite of Jordan blocks, then $ \begin{align} \det(Ax-I) & =\det(PJP^{-1}x-I)\\ &=\det(PJP^{-1}x-PIP^{-1})\\ &=\det(P)\det(Jx-I)\det(P^{-1})\\ &=\det(Jx-I) \end{align} $ so let's assume that $A$ is already in its Jordan canonical form. Some of the Jordan blocks of $A$ have eigenvalue $0$ and some do not. Write $A=\begin{bmatrix} Z & 0\\ 0 & Y \end{bmatrix} $ where $Z$ has the Jordan blocks with eigenvalue $0$, and $Y$ has the other (nonzero eigenvalued) Jordan blocks. It's important to understand that $Z$ has $0$'s everywhere except for some $1$s at selected places along the $+1$-off-diagonal. Then $ \begin{align} \det(Ax-I)&=\det\left(\begin{bmatrix}Zx & 0\\0 & Yx\end{bmatrix}-I\right)\\ &=\det(Zx-I_{z\times z})\det(Yx-I_{y\times y})\\ &=(-1)^z\det(Yx-I_{y\times y})\\ &=(-1)^zp(x;Y) \end{align} $ where $z$ is the multiplicity of $0$ as an eigenvalue of $A$ and $y$ is the complement: $n-z$.
Since $Y$ is invertible, Alex Becker's answer can be applied. In summary: $\begin{align}p(x;A)&=(-1)^zp(x;Y)\\&=(-1)^z(-1)^y\det(Y)c(x;Y^{-1})\\&=(-1)^n\det(Y)c(x;Y^{-1})\end{align}$ That is, $p(x;A)$ is a certain multiple of the characteristic polynomial of $Y^{-1}$, where $Y$ is the composite of $A$'s invertible Jordan blocks. Said one more way, $p$ is a polynomial whose roots are the inverses of $A$'s nonzero eigenvalues, and the multiplicities are respected.