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.
Were toward Eternity – (24)
If we were at all uncertain that the speaker was living forever in her new life, the last line really seals it. She tells us that her first feeling about the horses has been confirmed. Talk about a woman with spot-on intuition.