Welcome to CS.SE :)
Suppose $G$ is a forest. Then so is every subgraph $G[X]$ induced by some $X \subseteq V(G)$, and clearly every forest has a node of degree at most one (either an isolated vertex or some leaf).
Now suppose that $G$ is a graph such that every subgraph $G[X]$ induced by some $X \subseteq V(G)$ has a vertex of degree at most one.
Then $G$ cannot contain a cycle, otherwise the vertices of said cycle would induce a subgraph where every vertex has degree at least $2$ (in a cycle, every vertex is adjacent to its two neighbors).
As $G$ is acyclical, it must be a forest.
Putting the two pieces together, we get your desired result.