For the ${\mathrm{S}}_{\mathrm{N}}2$ reaction of $\mathrm{CH}{}_3\mathrm{Br}$ with $\mathrm{NaN}{}_3$ at the same concentrations, the rate is highest in

Protic solvents (water, methanol, ethanol) form hydrogen bonds to the azide anion and lower its energy and reactivity. DMF is polar aprotic, dissolves the salt by solvating $\mathrm{Na}{}^{+}$ but leaves $\mathrm{N}{}_3{}^{-}$ poorly solvated, so the rate is highest in DMF.