For any student $v$, let $S(v)$ denote the set of teams that contain $v$. It is given that $S(v)$ is never empty. The other condition can be written as: for any $u,v$ who are friends, there exists $w$ where $S(u)\cap S(v)\supseteq S(w)$. For any fixed $u,v$, consider the specific $w$ such that $|S(w)|$ is minimal (denoted $w'$). Then if $x$ is $w'$'s friend, $S(w')\subseteq S(x)$ (otherwise, $S(w')\supset S(w')\cap S(x) \supseteq S(w'')$ for some $w''$, violating the minimality of $|S(w')|$). Simply pick any $T\in S(w')$, then it follows that $x\in T$ so $T$ is a "team with the center" (with $w'$ being the center). $T$ also contains $u$ and $v$, hence the claim is proven.