Why does the pressure exerted by a gas at a constant temperature decrease when the volume of the gas is increased?

Because for a given quantity of gas at constant temperature, ${P}_{1} {V}_{1} = {P}_{2} {V}_{2}$
And thus, ${P}_{2} = {P}_{1} {V}_{1} / {V}_{2}$
Since by definition of the problem ${V}_{1} / {V}_{2} < 1$, because ${V}_{2}$ has INCREASED with respect to ${V}_{1}$, ${P}_{2}$, is less than ${P}_{1}$, the initial pressure.