Before the rise of dating in the 1920s, potential couples courted instead and were accompanied by a chaperone to make sure they did not engage in improper behavior. Courting usually involved the male in question paying a call on the lady he was interested in at her home or taking her out for a drive in a carriage. In the context of the poem "Because I could not stop for Death—," Death and the speaker are the courting couple going for a carriage ride, while Immortality is the chaperone.
Combining a common courting ritual with Death as the wooing male gives the poem a gothic vibe. Gothic literature often blended the morbid with the desirable (sex and death being a common pair). By presenting Death as the speaker's suitor, the poem is giving death an almost romantic air, something to be desired and not feared. Arguably, this could be ominous as well. Some have argued Death is instead a seducer luring the speaker to her grave, his "kindly" stopping only a pretense, but considering the courteous manner in which Death is presented and the gentle tone of the poem, the more likely answer is that Death here is a gentleman caller rather than an unscrupulous cad.