Why is it possible to assume a contradiction in a disjunctive syllogism? I was reading about the principle of explosion with this example:

Assume two contradictory premises: A.) 'All ice cream is frozen.'; B.)
  'Not all ice cream is frozen.' Now, just to show that it's possible,
  say one wants to use those two premises to prove that: C.) 'Words
  don't exist'.
To do so, construct a disjunction out of A and C: 'All ice cream is
  frozen or words don't exist.'
This statement appears to be perfectly acceptable here because it
  holds true under any of these three circumstances:
  1. All ice cream is frozen.
  2. Words don't exist.
  3. All ice cream is frozen and words don't exist. (Of which at least the first one is true because it was assumed as a premise.)
Now use that disjunction for a disjunctive syllogism: 'All ice cream
  is frozen or words don't exist. Not all ice cream is frozen. Therefore
  words don't exist.'
This also appears to be perfectly acceptable here because if it is
  said that at least one of A or C are true, then when it turns out A is
  not true (which is B, which has been accepted as a premise), at least
  it can be held that C is true.

However, the only issue I have with this is we assumed B is true at the beginning. So if B is true, then how are we sure that C is true? Aren't we just returning back to the original disjunction, but now saying "not all ice cream is frozen or words don't exist"? Or am I misinterpreting when they say "assume two contradictory premises" - I'm not sure if they mean to assume they're both true or not. I can see how a disjunctive syllogism works without contradictory assumptions, but now I'm confused on how this logic can hold true. 
 A: This looks badly written. I would have changed it to:
Assume two contradictory premises: A.) 'All ice cream is frozen.'; B.) 'Not all ice cream is frozen.' 
[Note: If this is confusing, that's sort of the point. There is nothing wrong with accepting as true a statement and its negation, because you're not aware of the consequences of what you assume in general. The logical consequence of assuming A and B, however, is that you can prove anything as a result, which is seriously wrong. It's not just a bad feeling; it's actual logic.]
Now, just to show that it's possible, say one wants to use those two premises to prove that: C.) 'Words don't exist'.
[That "it" must refer to something in the book before what you quoted. I would guess that "it" is something like "proving anything, under the assumption that you're assuming that two contradictory things are true".]
To do so, construct a disjunction out of A and C: 'All ice cream is frozen or words don't exist.'
[so far, so good. Then I would have continued with:]
Now, "A or C" is true, because we're assuming that A is true, and true "or'ed" with any value is true as well.
[Nothing should have been said about whether C is true or false, because we shouldn't be assuming anything about it at this point. The goal is to show that C MUST be true. Back to the original]
Now use that disjunction for a disjunctive syllogism: 'All ice cream is frozen or words don't exist. Not all ice cream is frozen. Therefore words don't exist.'
[end of rewrite]
A: We have to remind the definition of valid argument :

An argument from given premisses to a particular conclusion is valid if and only if there is no possible situation in which the premisses would be true and the conclusion false.

If we apply it to your case :

$$A \lor \lnot C, \lnot A \vdash \lnot C$$

we have that if $A$ is true, then $\lnot A$ is false; thus, in order to satisfy the second premise : $\lnot A$, we are left with only one possibility : $A$ must be false.
If $A$ is false, in order to satisfy the first premise : $A \lor \lnot C$ (see truth table for $\lor$) we must have $\lnot C$ true.
Thus the conclusion is entailed by the premises and the argument is valid :

if the premises are true, then the conclusion must be true also.


When we "apply" the above argument form to a "real life" example, it stay valid, but it licenses us to assert the truth of the conclusion only if the premises are both true.
In your case, if we agree that "All ice cream is frozen" ($A$) is true, we have that the disjuncttion "All ice cream is frozen or words don't exist" ($A \lor \lnot C$) is also true, but the second premise : "Not all ice cream is frozen"  ($\lnot A$) must be false, and thus we cannot conclude.
A: The issue at hand is that the conclusion of a valid argument can be nonsense.
Quite so.   An argument is sound only if both the form is valid and the premises are justified.
Here, you have a valid form, the disjunctive syllogism.   However, the premises are quite unjustified.   So it's not sound.
