Recall the definition of $s$:
$$s=(p_a+p_b)^2 = p_a^2 + p_b^2 +2p_a\cdot p_b$$
Where $p_a$ and $p_b$ are the 4-momenta of each colliding proton. For head on collision of particles with same energy and momentum as is the case with proton-proton collisions
$p_a=(E_p,\underline{p})$ and $p_b=(E_p,-\underline{p})$
Where $E_p$ is the energy of the proton and $E_p$>> the protons mass, $m_p$
$$s=(E_p, \underline{p})^2+ (E_p, -\underline{p})^2+2(E_p^2, \underline{p})\cdot(E_p^2, -\underline{p})$$
$$s=E_p^2+ |\underline{p}|^2+ E_p^2+|\underline{p}|^2+2E_p^2 -2|\underline{p}|^2$$
Sum together to get
\begin{eqnarray}
s=& 4E_p^2
\end{eqnarray}
The square root of $s$ gives the energy at the centre of mass
$$\sqrt{s}=2\sqrt{E_p}$$