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How do branching chain reactions lead to explosions involving hydrogen and oxygen?

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dissociation of E–N — reaction mechanism Possible energy diagram for the dissociation of a covalent molecule, E–N, into its ions E⁺ and N⁻ (see text).

In a branching chain reaction, the number of reactive intermediates—known as chain carriers—increases with each propagation step, causing the reaction rate to accelerate exponentially and resulting in an explosion.12

In the specific case of a hydrogen-oxygen explosion, this process is driven by the rapid multiplication of free radicals. A key step in this mechanism is the reaction between a hydrogen atom and an oxygen molecule:2
$H + O_{2} \to OH + O$
In this single step, one chain carrier (the hydrogen atom) produces two new chain carriers (a hydroxyl radical and an oxygen atom). Because each of these new intermediates can go on to react with further hydrogen or oxygen molecules to generate even more carriers, the concentration of these reactive species rises to a high value almost instantaneously.2

This uncontrolled increase in chain carriers prevents the reaction from reaching a steady state, causing it to proceed with extreme speed—often completing in less than 1/1,000th of a second—which is characterized as a chemical explosion.12