The Pigeonhole Principle states that if there are $(N+1)$ pigeons to fit in $N$ holes, at least one hole will have $2$ or more pigeons. Hence, if you pick $3$ socks (pigeons) from $2$ colors (holes), at least one color (hole) will have $2$ or more socks (pigeons), i.e. a pair is guaranteed with either red or black color.

In general, if there are $N$ different colored socks, then we would draw $N+1$ socks to ensure at least one matching pair.