Here is a sketch of a proof:  Let $\dot f$ be a name for $f$.  Wlog we may assume that 
$\dot f$ is a nice name and hence uses only countably many conditions.
It follows that it is an $Fn(J,2)$-name for some countable set $J\subseteq I$.
Let $(p_n)_{n\in\omega}$ be an enumeration of $Fn(J,2)$.

We construct a function $g\in M$ as follows.
For each $n\in\omega$ choose $g(n)$ such that for all $k\leq n$ the following holds:
if $p_k\Vdash\dot f(n)=m$ for some $m\in\omega$, then $g(n)\geq m$.