Edit: I was going to spoiler tag this in case people didn't finish the game.. anyway rewording the entire thing.
There are two main theories about the meaning of the ending. One focuses on what occurs at the ending using words and what actually happens. The second is more about the wording of the song itself.
The song itself is called Cara Mia Addio.
The lyrics translated in English.
Dear beautiful, my beautiful darling!
My child, oh Chell! [pun: "oh heavens!"]
For I hold her in esteem...
For I hold her in esteem.
Farewell to my dear!
My dear girl,
why do you not walk away?
Yes, away from Science,
My dear, my dear girl?
Ah, my beautiful!
Ah, my dear!
Ah, my dear!
Ah, my child!
Oh dear, my dear...
These are the theories found at the wikia http://half-life.wikia.com/wiki/Turret_Opera :
The first one is thought to be that she had died in the elevator. This theory is because the song being sung is like she is being sent to Heaven. In Greek/Roman Mythology when someone dies they arrive at a field with their closest companion. The field in Greek is known as Elysium.
When Chell arrives to see the turret quartet in the elevator, they
begin singing. Then after arriving at a wheatfield, Chell is
accompanied by the Companion Cube, her only friend. Chell is believed
(in this theory) to have died in the elevator with the 4 turrets."
The second main theory is that Chell is actually the offspring of both Cave Johnson and Caroline. This theory is due to the fact that the song as translated earlier hints at this. The meaning of the words could be interpreted this way because a Mother would call their daughter darling and dear.
This song could also be used to prove the theory that Chell is
daughter of Cave Johnson and Caroline. If you think that GLaDOS was
singing the lyrics in English there are words like "Darling" and
"Dear", words a mother would use to call her daughter.
Another potential nod to the fact that Chell could be the daughter was mentioned in an IGN Article Portal 2 Ending in 2011.
First off, we should assume that Caroline is the biological mother of
Chell, as evident by the turret song and several of GlaDOS' lines
towards the end. Not to mention that Chell literally means 'daughter.'
Now these theories are fun but Valve has never confirmed that these theories are in fact true. The only thing that we know is that Valve hints at the fact that Chell is actually alive in the DLC Peer Review as well as in the Co-Op Campaign.
There is no official corroboration from Valve, or any evidence of this
being the case. In addition, since the protagonist will often die
while players go through the game (prior to reattempting the stage
they died in), the occurrence of Chell's death is very familiar to
most, if not all, players, and a similar experience is notably absent
from the final elevator scene, casting doubt on any such
interpretations. Furthermore, GLaDOS seems to refer to Chell during
the Peer Review DLC, fearing that an old prototype of her chassis has
been taken over by a "mystery woman" and that "she" has returned
(believing that "she" is behind many of the malfunctions that occur
throughout the game, unaware that the source of the malfunctions is a
bird) where "she" could be Chell, or Caroline if it's wrong.