 Solution

If $G$ is $n$-transitive, it is transitive by extending one-element tuples to $n$-tuples. Apply $n$-transitivity to tuples beginning with $\omega$ to see that the stabilizer subgroup $G_{\omega }$ is $(n-1)$-transitive on the remaining elements.

Conversely, given two ordered $n$-tuples of distinct elements, transitivity first supplies $g$ carrying their first entries to each other. Conjugating the given stabilizer hypothesis shows that the stabilizer of the target first entry is $(n-1)$-transitive. An element of that stabilizer then maps the remaining entries of the image tuple to their targets. Composing the two elements proves $n$-transitivity. For $n=1$, the condition on zero-tuples is vacuous.