There's some info about this posted in the "Why are child pilots used" question, but specifically, Asuka's mom Kyoko has always been in the Evangelion.
In episode 22, we see the flashback of Asuka's mom having gone through a contact experiment similar to what Shinji's mom, Yui, went through with Unit-01. Similar to how Shinji's mom is "inside" Unit-01, Asuka's mom is "inside" Unit-02. I've put "inside" in quotes as it's probably not in a literal sense. Maybe it's their "souls" or "impressions" or something that's inside the Evangelions. It's not entirely clear why this mother-child bond is required in the show, although we do know the idea came from Sadamoto, the original mangaka. We also know it's somewhat an important functional requirement, until Gendo (and eventually Seele) managed to get the Dummy Plugs to work (e.g. Unit-03, Mass Produced Evangelion series from End of Evangelion).
As far as the "meaning of the A.T. Field", there's a good bit of speculation on how that all works. The first hint is from Kaworu during the end of episode 24 where he does a rather cryptic infodump on the mechanics of Evangelion, souls, and A.T. Fields:
Kaworu: Eva is made of the same body as me.
Because I'm also born of Adam.
When the unit doesn't have a soul, I can unite with it.
The soul of Unit 02 is shutting itself up now.
Shinji: AT Field!
Kaworu: Yes, you Lilims call it that.
The holy region that must not be invaded by anyone.
The light of the soul.
You Lilims are aware of that.
Aware that the AT Field is the wall of the soul that everyone
So A.T. Fields are generated by a "soul" (again, probably not something that is literal), and a "soul" is inside every Evangelion, and in this case, specifically Unit-02, Asuka's Evangelion. I you also recall, Unit-02 was previously unresponsive after the immense psychic attack from Arael which affected both Asuka and the Evangelion (e.g. her mom). This made it possible for Kaworu to take control of Unit-02 during episode 24 and why he says that the soul has "shut itself up". Asuka has come to the realization that:
- Her mom has been in the Evangelion this whole time
- Her mom issues were all for naught because her mom has been with her this whole time, she wasn't alone afterall
- That her mom was using her soul to create Unit-02's A.T. Field which was protecting Asuka
At that point, Asuka goes into kick-ass mode with/because of her new understanding; a renewed bond with her mother/Eva's soul, a new understanding of what the A.T. Field really is and probably an enhanced control of how to use it (as evidenced by her fight with the Mass Produced Evangelions)