The holographic principle is an idea that all of the information about three dimensional volume can extracted by measuring the two dimensional boundary of the region. This is a very deep (and speculative) concept in physics that is motivated by black hole thermodynamics, general relativity, and string theory. There are concrete examples of the holographic principle in string theory, unfortunately none of them are simple. But if you want to learn more you might start reading a little here (Holographic principle AdS/CFT).

The anthropic principle states that even if the odds are very small for intelligent life to exist, we still can't properly call our existence a coincidence, since we can only ask the question "does life exist?" if we already exist. It's like asking the question "What are the chances that I was born?" If you think about it carefully, the correct answer is 100%.

I cannot explain the holographic principle; it appears to be something that is consistent with the theory of thermodynamics, and is used to explain why we see the universe in three dimensions when, according to equations, black holes behave thermodynamically as though there were only two dimensions. I should warn you that string theory (which the holographic principle is part of) is an untested and untestable area of physics and as a result most physicists are highly skeptical of it, even going so far as to call it unscientific for being untestable.

The anthropic principle is a principle that the area of the universe that we see is the way we see it because otherwise we couldn't be here to observe it. For example, we know now of many many different solar systems with planets, but very few of them, except our own, are suitable to have a habitable planet like Earth. This means that planets with oceans of liquid water may in fact be very rare in the universe. We happen to live on one because there is nowhere else that we could have evolved.