What does this paraphrase of the birthday problem mean? The following is an excerpt from A Generalized Birthday Problem - David Wagner:


  One of the best-known combinatorial tools in cryptology is the birthday problem:
  Problem 1. Given two lists $L_1, \space L_2$ of elements drawn uniformly and independently at random from $\{0, 1\}^n$, find $x_1 \in L_1$ and $x_2 \in L_2$ such that $x_1 \oplus x_2 = 0$.


It's not so intuitive for me to understand. In my understanding, the birthday problem is about the probability that at least 2 people in a room have the same birthday. How does the birthday problem transfers to this? Please give me some hints.
$x_1 \oplus x_2 = 0$ 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.