What is a false premise? I don't understand what a sound argument is. And what does it mean for a premise to be false? Why does case 3 (A is false, B is true) not apply in the real world?
Here the author says that the first premise is false. But how can $A \lor B$ be false in case 3 when $B$ is true?

 A: Note the very first sentence in the last paragraph: "Viewed as atomic sentences". With this, you're supposed to strip the sentences of their meaning and look at them just as literals (i.e. propositional letters or negations of these).
Thus you can attribute truth values to the propositional letters, (see valuations).
Then the author talks about case 3 which I'm guessing is the valuation $(A,B)\mapsto (F,T)$.
 This valuation makes both the premises true and the conclusion true also.
Now the author says that in the real world case 3 can't happen, this means that if you now stop looking at the sentences as propositional letters and give them meaning again, then something impossible happens, namely that Teller has never taught logic (this is somehow known to be false, maybe Teller is the author).
The term false is being used with two different meanings, one of them is the natural language interpretation, the other one is as a valuation.
A: Using the rules of natural deduction or a truth table, you can prove that $$[A\lor B]\land \neg A \implies B$$
As such, it is a valid argument.
It would be a sound argument if and only if both premises $[A\lor B]$ and $\neg A$ are true. 
It would be unsound if and only if either premise is false.
A: The following proof is a valid argument, however, the conclusion of the theorem is clearly false. What went wrong?

Theorem 1 Let $1 = 0$, then all natural numbers are equal to zero.
Proof by induction. Obviously, $0 = 0$. Now, let $k$ be any natural number $\geq 1$. By inductive hypothesis we have $k-1 = 0$. Using our assumption we get $k-1+1 = 0+0$, that is $k = 0$ which concludes the proof.

Some funny examples of this kind happen with loaded questions. For example, if you were to answer the well-known loaded question presented below by "Yes, I have" or "No, I haven't",
                                         Have you stopped beating your wife?
then you would admit that, at some point, you were doing it (and that you actually have a wife). To respond to such a question, one usually points out (in whatever way) that it is based on false premises.
A sound argument is one which is both valid and its premises are true. The above is not sound, because the premise $0 = 1$ is not true. Still, the difference is rather subtle. For example, if the conclusion of the theorem was the whole implication $$(0=1) \implies \forall k\in\mathbb{N}.\ k=0$$ then the only premises are the axioms of logic, natural numbers, etc., and such a theorem would be both valid and sound.
I hope this helps $\ddot\smile$
A: I recently answered this question about valid and invalid arguments, see if it helps.
An argument is sound if it is valid and has true premises.  In this case the conclusion is guaranteed true.  I don't think your author's example is very good.  See if this one helps.

Anyone born outside the USA cannot legally be president.
Barack Obama was born outside the USA.
Therefore, Barack Obama cannot legally be president.

I hope you agree that the logic here is perfectly correct: if the first two statements are true, then the conclusion has to be true.  The problem is that (according to most people) the first two statements are not both true - specifically, the second is false - and therefore the conclusion is not guaranteed to be true.  This is an example of an argument which is valid but not sound.
A: A sound argument comes as one where the premises are true, and the conclusion is true.  For example:
(2+2)=4
Therefore,
(2+2)=(4+0)
makes for a sound argument as does:
(4+6)=10
(5+1)=6
Therefore,
(4+(5+1))=10
Case 3 doesn't hold, because neither does it hold that Teller is ten feet tall, nor does it hold that he hasn't taught logic.
One reason the notion of a valid argument becomes important, because we might end up studying or working with some system where we don't know whether the premises hold true, or if they hold false (but we can tell that one of the two must hold... or we know what the possibilities are).  For instance, suppose that we know we have some particular binary operation % given to us.  We're given the following argument
(3%4)=7
4=(1%3)
Therefore,
(3%(1%3))=7
The argument holds as valid no matter what "%" indicates, because of the structure involved here.  The premises could hold true... say if "%" indicates addition.  But, the premises could hold false also... say if "%" indicates multiplication.  Either way though, the argument holds valid by its form.  Consequently, we can tell something about "%" and know that certain consequences will follow even if we can't figure out the truth of the premises.
As perhaps a better example to illustrate why validity can be important, one can argue:
If a set of axioms A is independent, then there does not exist a model of the axioms which has some property P.
There does exist a model of the axioms which has some property P.
Therefore, the set of axiom A is not independent.
This argument holds as valid.  But we can't tell a priori whether or not for a given set of axioms there exists a model of the axioms which has some property P.  But, the argument holds as valid nonetheless and thus enables a sound argument, in a particular case, if we can find some instance where the premises hold true.
