Gödel's second incompleteness theorem states that if $\mathsf{ZF-Inf}$ is consistent, then $\mathsf{ZF-Inf} \nvdash \mathsf{Con(ZF-Inf)}$. Moreover, if $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$ were inconsistent, then (under the condition of $\mathsf{ZF-Inf}$ being consistent) in every model of $\mathsf{ZF-Inf}$ it is true that $\mathsf{Con(ZF-Inf)}$. Thus, $\mathsf{ZF-Inf} \vdash \mathsf{Con(ZF-Inf)}$. By contradiction, this implies that $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$ is consistent.
How can I imagine a model of $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$?
In my opinion it seems quite strange to have a model of $\mathsf{ZF-Inf}$ where $\mathsf{ZF-Inf}$ is not consistent. Does this mean that it is possible to prove everything in this model? This is not possible as $M \vDash\varphi$ and $M \vDash \neg \varphi$ are not possible for a model $M$.
Or, let $M$ be a model of $\mathsf{(ZF-Inf)} + \neg \mathsf{Con(ZF-Inf)}$. Then is every encoding of $\mathsf{ZF-Inf}$ in $M$ inconsistent, i.e. $PA$ and arithmetics are inconsistent in $M$. So $M$ is a model in which arithmetics is inconsistent?
Maybe this question isn't that clever or shows a lack of understanding, then please let me know.