Why does hydrogen burn spontaneously in fuel cell at room temperature? It's well known that if you mix $\ce{H2}$ and $\ce{O2}$ in ambient conditions, the $\ce{H2}$ will be stable indefinitely, and will only react with $\ce{O2}$ to form water when ignited by a flame. The reason being that for the reaction

$$\ce{2H2 + O2 -&gt; 2H2O} \label{rxn:1}\tag{1}$$

to occur, first the bonds must be broken in $\ce{H2}$ and in $\ce{O2},$ which is an insurmountable activation energy at room temperature.

So I am confused why a hydrogen fuel cell works at room temperature without ignition, as it is powered by the same chemical reaction \eqref{rxn:1}.

In particular, why can the strong double bond in $\ce{O2}$ suddenly be broken in this context without a flame?

My best guess is that somehow the electron sea in the copper wire acts as a catalyst. If this is true, what are the details of this mechanism that allows the $\ce{O2}$ bond to break at room temperature?
You guessed it right that hydrogen and oxygen mixtures will remain stable. Someone in Harvard waited for >30 years and found very little water if any. As you said, there is a energy barrier. What type of materials can lower this barrier? Catalysts.

The electrode materials require some catalysts such as platinum or nickel. Of course, making cheaper and sustainable catalysts is an extensive area of research. If a person discovers a better catalyst which is far more common than Pt, then that person certainly deserves a big award. Copper does not act as a catalyst here.