Another way to phrase this exercise is to say that any undirected graph with at least two vertices has at least a pair of vertices of the same degree -- because we can represent people with vertices and the relationship "i knows j" by an edge {i,j}.
A graph on n vertices has a choice of degrees for its vertices from a minimum of 0 to a maximum of n-1; thus, if no two vertices have the same degree, then the graph must have exactly one vertex of each degree. But that is impossible: since one vertex has degree n-1, it is connected to all other vertices, so that there cannot be a vertex of degree 0. Hence there must be at least one pair of vertices with the same degree.