Why does $p$ (is true) strictly agree with $p$ while $p$ (is false) strictly disagrees? Let's make the truth table:
$$\begin{array}{|c|c|c|}
\hline
p&(p) \text{ is true}&(p) \text{ is false}\\ \hline
T&T&F\\
F&F&T\\\hline
\end{array}$$
"$p$ is true" strictly agrees with $p$, while "$p$ is false" strictly disagrees with $p$. Why this asymmetry? 
It just seems to oddly line up with natural language, and I am wondering if it does so just because we set it up that way on purpose. What I mean is that the proposition $p$ strictly agrees with the assertion $p$. In natural language, when someone says "If it is sunny, I will go to the beach" we expect that they are asserting that their statement is true, not that it is false; If we were assuming that they were asserting it is False, we would be wrong. In the same way, we seem to have "conveniently" set up logic so that when we see a written proposition $p$ we can assume the statement to be "$p$ is true" without changing anything.
Is there something fundamental about the definition of $T$ and $F$, or an axiom, that makes "True" special?
 A: Let me rewrite my earlier answer completely since I think I understand a bit better what you are asking.
As I wrote above, you have to be careful when you start using mathematical logic in the real world. Often one hears statements like: I can mathematically prove that God exists or the opposite: I can mathematically prove that God does not exist. The problem here is that any mathematical proof involves only well defined terms. I might prove that all non-trivial proper subgroups of an Abelian group are normal. In this proof I would then use the definition of Abelian. I would use the definition of normal subgroup and a bunch of other things. All these terms have been defined elsewhere.
When I teach logic (low level) I also often give the examples like: Let $A$ be the statement that is is raining today. Then ... The problem here is that I haven't (mathematically) defined raining and I haven't defined today and so on. So this is actually a really bad example. We are trying to teach how mathematical logic is this precise way of doing mathematics, but then we star giving non-mathematical examples.
Now, I still do give these examples because one often can use ones intuition from the real world in logic. The basic task of negating a statement if many times done using your natural understanding of the words involved. 
But this can be dangerous. My favorite example is the use of the word "or". In the real world, this (often) means exclusive or, but in mathematics $A$ or $B$ is true also if both $A$ and $B$ are true.
So what can you do?
Here is a suggestion that has helped me: Don't worry about the real world equivalences of mathematical terms and objects. Don't overthink things. Just do back to the definition. As such, saying that $p$ is false, is just the definition of negating $p$. And so, by definition $p$ is false has the opposite truth value of $p$. 
Does that mean you can't/shouldn't ask about these things? No, not necessarily. But I would argue that the questions like "What is truth?" are more philosophical than mathematical.
A: Saying that a statement is true, is just saying the statement itself.
For instance, there's no real difference between me telling you 

Today is Wednesday

and

It is true that today is Wednesday.

Similarly, saying that a statement is false, is just saying the opposite of the statement. There's no difference between me telling you 

Today is not Wednesday

and

It is false that today is Wednesday.

So "is true" preserve truth values, and "is false" flips them.

I'm actually being a bit glib here: some philosophers of language question whether "$p$ is true" always has the same meaning as "$p$." For example, "This sentence has five words" is true, but "It is true that this sentence has five words" is false. I'd argue that that's a misleading example, and that when construed appropriately what I've written above is even true in natural language, but I don't want to get too deeply into that here. Remember that the propositions in formal logic are of a particularly tame kind, compared to those of natural language; so in this context (except for the further reaches of nonclassical logic), what I've written above is true.
A: A few suggestions in response to your request for books etc.
My favourite author in this area is Raymond Smullyan . He is sometimes thought of as a mere populariser, but he is much more than that. His books cover a wide range of difficulty. At one level we have his famous logic puzzle books, which are aimed at a fairly large market. Then we have his fairly straightforward introductions (like first order logic). But we also have his ContinuumProblem book which covers advanced material. I remember struggling to understand Paul Cohen's original book and finding it heavy going. Smullyan's book is far easier to follow.
For immediate reading, maybe one or more of (1) What is the name of this book? (2) A beginner's guide to Mathematical Logic, (3) The Godelian puzzle book, (4) To mock a Mockingbird, (5) King Arthur in Search or (6) First Order Logic. 
In passing I do not really agree with @Thomas ' comment above "I would simply take these things as definitions and try not to worry too much about their real world every day life language equivalent". I think it is extremely important to understand why definitions are the way they are. In general math based on arbitrary definitions tends to be sterile. For some mysterious reason, it usually works better (and seems "deeper") when it has some kind of contact with reality.
