The displacement reactions of the halogens are redox reactions, because oxidation and reduction happen together. Oxidation is loss of electrons and reduction is gain of electrons (remember OIL RIG: oxidation is loss, reduction is gain).
Take chlorine and potassium bromide. The ionic equation is Cl2 + 2Br- → 2Cl- + Br2. Each chlorine molecule gains two electrons to form two chloride ions: Cl2 + 2e- → 2Cl-. So chlorine is reduced. Each pair of bromide ions loses two electrons to form a bromine molecule: 2Br- → Br2 + 2e-. So bromide ions are oxidised.
The potassium ions are the same on both sides, so they do not change and are spectator ions. The same pattern works for chlorine with iodide ions, and bromine with iodide ions: the more reactive halogen is reduced and the halide ions are oxidised.