Although still not fully understood, the mechanism of extinction by halons can be explained by an effect on the radical reactions that take place during combustion.
The conversion of combustible material to carbon dioxide and water, the final stage of combustion, takes place by combination with oxygen in the air passes through intermediate stages involving free radicals.

RH + O2 = R + H0 + 0 + Ho
Comb mat’l + oxygen = free radicals

CF3 Br = CF3 + Bro
Br radical
HBr +OHo = H20 + Bro

The halons dissociate under the action of heat, giving bromine radicals
The Br radicals combine to give HBr, which is able to block the Hydroxl radical, Ho, regarded as being responsible for the development of fire. Halons 1211 & 1301 contain about 50% bromine, which is the key component in extinguishing a fore. This explains why only a small amount of halon in air (4%), is enough to extinguish a fire!