I solved this as a homework question with perturbation, or rather, the equivalent identity, $\sum_{k=0}^{n}{(k^2+1)k!}=n(n+1)!+1$.
The most promising idea I had was that the right hand side looks like an episode of a game show with $n+1$ contestants. They complete a series of tasks which results in a ranking. At the end of the episode the winner chooses one of the remaining $n$ contestants to be eliminated. The RHS counts the number of ways such an episode could transpire.
I cannot however find a good interpretation of the LHS. I tried looking at it as e.g. $\sum_{k=1}^{n}{({k\choose k-1}^2+{k \choose k}^2)k!}$, but I'm getting nowhere.