The question is essentially just the title.
Clarification: Here is an example with $n=3$ and $k=1$: $k$-set $\to$ $k+1$-set; $1\to12$, $2\to23$, $3\to13$.
Since $n>2k$, I know that there are enough $k+1$-sets for a mapping between the two to exist, but I am struggling to see how I know that the mapping ensures that the sets are extended.
I know that each $k$-set can be extended in $n-k$ different ways, but I don't know if it is true that there is a way of extending each $k$-set such that each set always has an available way to extend.
I was told that this problem uses Hall's Matching theorem, but that is a result from graph theory and I do not really know how to apply it here.
Consider the bipartite graph with vertices consisting of $k$-sets and $(k+1)$ sets, with an edge between $A$ (a $k$-set) and $B$ (a $(k+1)$-set) iff $A \subset B$. As you've said, each $k$-set $A$ has degree $n-k$. Can you now see how to use Hall?