Well, from an operational standpoint, you would connect the vacuum pump (or water aspirator, if that's what you're using) - via thick-walled rubber tubing, usually - to that "nipple" labeled "vacuum adaptor." Once connected (and assuming all the other joints are tight), the pump/aspirator will lower the air/vapor pressure inside the entire apparatus (essentially by sucking air out of it).
Liquids generally boil at a lower temperature when under lower pressure. The lower the pressure (i.e. stronger the vacuum), the lower the boiling point. So, if you apply a strong enough vacuum, something that normally boils at, say, 150 deg C under normal (atmospheric) pressure can be distilled at, for example, 70 deg C. With a super strong vacuum pump, you could probably lower the pressure enough to distill the same liquid at maybe 50 deg C (or lower).
Usually the reason distillations are done under vacuum is because its safer to distill things off at lower temperatures. Lower temperatures = fewer side-reactions and/or less decomposition than if you were to perform the distillation without applying vacuum. Things can get kinda messy if you heat a flask and its contents to 150-200 deg C for several hours.