Ah wait, it's much easier to see contrapositively. If $p$ is interior in $M$, every tangent vector at $p$ can be represented by a curve with domain $(-\epsilon,\epsilon)$, but then, since $f$ is a submersion, every tangent vector at $f(p)$ can be represented by a curve with domain $(-\epsilon,\epsilon)$ and so the image is an interior point.