How does the Holographic Principle help to establish the fact that all the information is not lost in a black hole?
As I said in the comment, the information is not lost simply because of the holographic image at event horizon. This was the result of a long 20 year battle between Susskind and Hawking. Susskind had the final laugh!
I'll quote an excerpt from wiki,
This idea was made more precise by Leonard Susskind, who had also been developing holography, largely independently. Susskind argued that the oscillation of the horizon of a black hole is a complete description of both the infalling and outgoing matter, because the world-sheet theory of string theory was just such a holographic description. While short strings have zero entropy, he could identify long highly excited string states with ordinary black holes. This was a deep advance because it revealed that strings have a classical interpretation in terms of black holes.
The Holographic principle is one manifestation of the ADS/CFT correspondence - it's mentioned in the Wikipedia article you cited. The correspondence means that the same black hole can be described using a 5D string theory and a 4D (supersymmetric) field theory. The two are just different ways of describing the same object. In the 5D string theory black holes look quite simple and it's obvious there is no information loss, so we can be confident that there is no information loss in the 4D description either.
So far so good, but no-one knows how to describe the information "no loss"in 4D. Susskind's proposal is possible, but I'm sure he would agree it's not a proof. If someone could work out exactly how the "no loss" in 5D is described in 4D that would be quit a step forward.
I would expect the resolution of this dilemma to hold no relation whatsoever to AdS/CFT, being it specific to anti de Sitter spaces, which makes it irrelevant to our universe (unless someone proves an analog version for de Sitter spaces, which seems unlikely)
see this answer for details about an alternative resolution.