Where in 5e does it say this?
Good question. It is one of those things I read "somewhere" probably from an earlier edition or a specific setting.
When combing thru the 5e DMs Guide, I found two different versions of Hades, one the default (64) and one for a nondefault Greek-esque campaign setting (44).
"In the Greek cosmology, ... in Hades, named for its ruler, mortal souls linger as insubstantial shades until they fade into nothing."
(This "Hades" is more like Shadowfell. These Greek afterlifes are still "in the Material Plane", including in the Elysian Fields, Hades, Tartarus, and a literal Mount Olympus.)
In the default afterlife, souls sort themselves by deity or if without a deity then by alignment. Any Evil souls that are unclaimed by a Fiend, "become larvae and spend eternity in this place" of "despair".
(Oppositely, Elysium is "a heaven of well earned rest" and "joy".)
My impression is, the default 5e afterlife is all souls remain somewhere in the Astral Plane, including one of the Outer Planes within it, for "eternity". Except irregular events can happen such as
Resurrection or so on.
These eternal souls dont age. Likewise, the disembodied souls that travel the Astral Plane, including the Outer Planes, are also eternal.
(Afterlife souls generally dont go on adventures, but probably could, whether by choice or exceptional circumstance.)