What does this paraphrase of the birthday problem mean?
$x_1 \oplus x_2 = 0$ is equivalent to $x_1=x_2$ (because $\oplus$ is bitwise XOR, and that equivalence stands for bits, and multibit quantities being equal in all their respective bits is equivalent to these quantities being equal).

Now assume that $x_i$ is the birthday of person $i$ in the room, expressed as days since the first day of the year, in binary, with a year of $2^n$ days, and what's meant should be clear.

Notice that the problem studied in the quote is about two lists/rooms, rather than one in the standard birthday problem.