The protagonist/speaker is, first of all, dead. She is greeted by Death, personified. Death arrives on a carriage to pick up the protagonist. Death functions here like a date or one who is seducing the protagonist. As if to put her mind at ease, Death has brought Immortality (also personified) with him to act as a chaperon. The protagonist is seduced or at the least, she feels comfortable during this journey with Death: 

We slowly drove--He knew no haste

And I had put away

My labor and leisure too,

For His Civility--

At a point during the fourth stanza, the protagonist senses a chill in the air, perhaps a first sense of foreboding in what had been, up to that point, a comfortable ride. Finally, the protagonist concludes that these centuries since they began their ride (when she first "surmised" she was headed for the grave - Eternity), have only felt like one day. In this eternity, she has a different sense of how time passes. The protagonist finds herself, in death, in a state where/when time has stopped; or, being in the eternal, time has become meaningless. This could imply a number of things. Consider that the ride itself lasts forever. Throughout all of this, the protagonist speaker seems to take it in stride; it's as if death, in being eternal, is an endless (centuries) but also a fleeting ("shorter than the Day") event. Therefore, the protagonist is simply trying to understand what death is exactly. The ride gives her a chance to do this.