Let $N:=\textrm{Neigh}_{Nis}(U,\nu)$$N:=\textrm{Neigh}\_{Nis}(U,\nu)$ denote the filtered category of Nisnevich neighborhoods of the point $\nu\in U$. Then $\textrm{colim}_{V\in N^{op}} V\cong \textrm{spec}(\mathcal{O}^{h}_{U,\nu})$$\textrm{colim}\_{V\in N^{op}} V\cong \textrm{spec}(\mathcal{O}^{h}\_{U,\nu})$, where $N^{op}$ denotes the opposite category of $N$ and $\mathcal{O}^{h}_{U,\nu}$$\mathcal{O}^{h}\_{U,\nu}$ is the henselization of the local ring $\mathcal{O}_{U,\nu}$$\mathcal{O}\_{U,\nu}$. My guess now would be that the fiber is the constant simplicial set associated to the set $\textrm{Hom}_{k}(\textrm{spec}(\mathcal{O}^{h}_{U,\nu}),X)$$\textrm{Hom}\_{k}(\textrm{spec}(\mathcal{O}^{h}\_{U,\nu}),X)$.
Maybe someone can clarify this as well as your second question.