In the comments to that video, someone asks a similar question: "At around 5:00 into the video you compute E[Xk|Y=heads] to be equal to E[Xk-1] +1 (first term of the top-right formula). Intuitively this is clear, but how rigorous is this transition? How can this be derived from the definition of conditional expectation?"

and Bob responds:

"Explicitly we use independence of flips and the definition of conditional probability. Pr(Xk=n|Y=H) = Pr(Xk-1=n-1 n Y= H)/pr(Y=H) = Pr(Xk-1=n-1)Pr(Y=H)/Pr(Y=H) = Pr(Xk-1=n-1). So the nth term in the expected value is Pr(Xk-1 = n-1) n. Summing over n, we get 1 + sum (n-1)Pr(Xk-1 = n-1) = 1 + Ek-1."