What happens at this part in the key schedule of a Speck...?
For example, in the key schedule in the link, the result of the binary addition is being XOR d with 0?


It's very simple: In the key schedule, the exclusive-or step after the addition is done with the round counter, and the round counter starts at $0$ rather than $1$.

So for round number $0$, the first exclusive-or is done with the value $0$. In the next round, the exclusive-or is done with the value $1$, and so forth.

The supplied reference code might be easier to read than the block diagram, depending on your programming skills.