To prove the identity $$\sum_{k=0}^n\frac{n!}{n^kk!(n-k)!}=\frac{(n+1)^n}{n^n}$$ by induction, the induction hypothesis is stated as $$\sum_{k=0}^n\left({n \choose k}\frac{1}{n^k}\right)=\left(1+\frac{1}{n}\right)^n$$, which relates to the binomial theorem. The base case for n=1 is verified as true, showing that $$\sum_{k=0}^1\left({1 \choose k}\frac{1}{1^k}\right)=2$$ matches the expected result. The next step involves restating the induction hypothesis for n and determining the appropriate induction step to prove the case for n+1. The discussion emphasizes the importance of using binomial coefficients and identities in the proof process.