 Solution

Fix a finite nonempty vertex set $K$. Once $G_n$ contains $K$ and every edge incident to it, every spanning tree of $G_n$ contains an edge of the finite cutbecause otherwise $K$ is disconnected from the rest of $G_n$. Henceand the same holds in the weak limit. If the free spanning forest had a finite component, its vertex set would be some finite connected $K$ and all edges of ${\partial }_{E}K$ would be absent. Taking the countable union over finite $K$ proves that every component is infinite almost surely.