I think you are mixing up pressure and vapour pressure. A liquid boils when its vapour pressure is equal to the pressure over the liquid. So, when the pressure outside the liquid is lowered, the boiling point is reached at a lower vapour pressure as well.
The vapour pressure increases with temperature, so when a lower vapour pressure is required for boiling it will occur at lower temperature.
A bit too many vapour pressures in there, but I hope it is clear enough