Feynman's random walk (6-3)

How does Feynman get to $D^2_{N-1}$?

The expected value of $D^2_N$ for $N>1$ can be obtained from $D_{N−1}$. If, after $N−1$ steps, we have $D_{N−1}$, then after $N$ steps we have $D_N=D_{N−1}+1$ or $D_N=D_{N−1}−1$.
And could someone help me with this sentence?

In a number of independent sequences, we expect to obtain each value one-half of the time, so our average expectation is just the average of the two possible values. The expected value of $D^2_N$ is then $D^2_{N−1}+1$. In general, we should expect for $D^2_{N−1}$ its “expected value” $⟨D^2_{N−1}⟩$ (by definition!).

https://www.feynmanlectures.caltech.edu/I_06.html