The Carriage held but just Ourselves – And Immortality. (3-4)
The speaker knows this journey to death is also the beginning of the afterlife. So she believes she will continue a life, just not here on Earth. No wonder she's not freaking out.
We passed the School, where Children strove At Recess – in the Ring – We passed the Fields of Gazing Grain – (9-11)
Children grow up and fields go dry – these are examples of things that won't last forever. As the speaker passes them, perhaps their transience reminds her of her soon-to-be immortality.
Since then – 'tis Centuries – […] (21)
Well, she's talking to us from somewhere, and it sure isn't Earth. We now have proof that she continued on somewhere after she died. "Centuries" is a really long time, so whatever afterlife it is (Heaven?) it's not a normal human life span, and we get a sense she'll live forever where she is now.