Oxidation of metals/halogens by oxygen gas in acidic aqueous solution
As noted in your question, the standard reduction potentials are given as such in tables. 


  $$
\begin{align}
\ce{O2(g) + 4 H+(aq) + 4 e- &amp;→ 2 H2O(l)} &amp;\quad E^\circ &amp;= \pu{+1.23 V} \\
\ce{Ag+ + e- &amp;→ Ag(s)} &amp;\quad E^\circ &amp;= \pu{+0.799 V} \\
\ce{Br2(l) + 2 e- &amp;→ 2 Br-(aq)} &amp;\quad E^\circ &amp;= \pu{+1.065 V}
\end{align}
$$


To get the the standard oxidation potentials you reverse the products and reactants and you must also flip the sign of the reaction. So:


  $$
\begin{align}
\ce{Ag(s) &amp;→ Ag+ + e-  } &amp;\quad E^\circ &amp;= \pu{-0.799 V} \\
\ce{2 Br-(aq) &amp;→ Br2(l) + 2 e- } &amp;\quad E^\circ &amp;= \pu{-1.065 V}
\end{align}
$$


Now there are two possible overall reactions (ignoring the Nernst equation):


  $$
\begin{align}
\ce{O2(g) + 4 H+(aq) + 4 e- &amp;→ 2 H2O(l)} &amp;\quad E^\circ &amp;= \pu{+1.23 V} \\
\ce{Ag(s) &amp;→ Ag+ + e-  } &amp;\quad E^\circ &amp;= \pu{-0.799 V} \\
\ce{O2(g) + 4 H+(aq) + 4 Ag(s)&amp;→ 4Ag+ + 2 H2O(l)} &amp;\quad E_\text{Total}^\circ &amp;= \pu{+1.23 - 0.799 = +0.43 V}
\end{align}
$$


and the other overall reaction is 


  $$
\begin{align}
\ce{O2(g) + 4 H+(aq) + 4 e- &amp;→ 2 H2O(l)} &amp;\quad E^\circ &amp;= \pu{+1.23 V} \\
\ce{2 Br-(aq) &amp;→ Br2(l) + 2 e- } &amp;\quad E^\circ &amp;= \pu{-1.065 V} \\
\ce{O2(g) + 4 H+(aq) + 4 Br-(aq) &amp;→ 2Br2(l) + 2 H2O(l)} &amp;\quad E_\text{Total}^\circ &amp;= \pu{+1.23 - 1.065 = +0.16 V}
\end{align}
$$


Since the EMF for both overall reactions is positive, the reactions should be spontaneous. If the overall EMF was negative then the reaction would proceed in the opposite direction (in other words, swap the products and reactants.) 

Notes:

(1) I don't know why the silver reaction doesn't happen. The EMF for the cell indicates that silver should oxidize in these conditions. 

(2) The EMF's calculated ignore the Nernst equation.

(3) The oxygen half cell reaction is a bit weird. Obviously the oxygen must be dissolved in the aqueous solution to react. I'm assuming that the notation is trying to point out that the dissolved oxygen is in equilibrium with the atmospheric oxygen which has a constant concentration which is inexhaustible.