What does ashes to ashes, dust to dust mean?

Probably has to do with getting cremated after you die.

Ha, ha, ha! No it is part of the Anglican (Church of England, Episcopalian) burial service: "ashes to ashes, dust to dust". Ashes and dust are synonymous terms which mean "dirt". (This is why "ashcan" means the same as "dustbin") The sense is that we start out as dirt (see the book of Genesis for the story of Adam being created from dirt) and we end up as dirt. Our physical existance is just a passing phase, so we should not grieve overmuch over the death of loved ones.