Why does electrolysis of aqueous concentrated sodium bromide produce bromine at the anode?
There is important to know the actual potential depends on actual activities of reagents ( approximately concentrations for dilute solutions).

Additionally, the electrode potential is the thermodynamic quantity. If some net reaction kinetic is involved, other potentials affects the actual potential at the solution border, like the diffusion potential and dielectric layer potential.

We consider reactions
$$\begin{align}
\ce{Br2 + 2e- &amp;&lt;=&gt; 2 Br-}\\
\ce{O2 + 2 H2O + 4e- &amp;&lt;=&gt; 4 OH-}\\
\end{align}$$

The initial bromine activity is very low.

Activity/concentration ratio of ions typically decreases with concentration, as seen in the Debye-Huckel equation,
but for high salt concentrations it often jumps up the roof. 

Therefore, activity of bromides is very high and the actual potential gets low.

$$E_{c_{\ce{Br2}}/c_{\ce{Br-}}}=E^{\circ}_{c_{\ce{Br2}}/c_{\ce{Br-}}} + 0.059 \log \left( \frac{a_{\ce{Br2}}}{a_{\ce{Br-}}}\right)$$

OTOH, 
The potential for hydroxide oxidation  is pushed up for neutral solutions to the middle between standard redox potentials for acidic and alkaline solutions $$(+0.401 + +1.229)/2=+0.815$$

$$E_{c_{\ce{O2}}/c_{\ce{OH-}}}=E^{\circ}_{c_{\ce{O2}}/c_{\ce{OH-}}} + 0.059\log \left( \frac{a_{\ce{O2}}}{a_{\ce{OH-}}}\right)$$



Very important thing is the oxygen overpotential. Its value is listed in range +0.56 V on Ni to +1.02 V on Au, in contrary to the maximal value for chlorine ( on graphite)  +0.12 V. 



For $\ce{NaCl}$ solutions, difference of standard potentials is too high to produce metallic sodium, unless one uses mercury as the cathode, where happens the hydrogen, related to its reaction kinetics. This is used for production of sodium hydroxide, where sodium amalgam reacts with water.