To break it down into standard boolean algebra, the statement "if A then B" is logically equivalent to saying "not A or B". !A|B = B|!A = "If not B then not A".
Now back to the paradox, using boolean terms. The statement can be rephrased as !A|B: "This sentence is false or Santa Claus exists." Since we know that Santa doesn't exist, the sentence must be false: !(!A|B). That is equivalent to A&!B: "This sentence is true and Santa Claus doesn't exist." Here is where we have the paradox. We just said that the sentence must be false, but that then implies that the sentence is true.
The main mind-bender here is to realize that the negation of "If this sentence is true, then Santa Claus exists." is "This sentence is true and Santa Claus doesn't exist." We are not able to get rid of the statement "this sentence is true" no matter which way we toggle it.
See the  link?
Seriously though, I thought about it, but then I realized that I would probably have to replace the entire paragraph, since it seems to be a key step in the original author's explanation.
Maybe I will replace it anyway, since it can't be a very good explanation if it relies on faulty logic.
Update: Did it. Now I'm contradicting myself. Gotta' love these logic games.
Basically assume A is true then the non existence of Santa Clause is a contradiction so A is false.
If (A then B) says nothing if A is false so your done.
I think the problem is in the acceptance of the if unconditionally without merit. That automatically couples it to the condition of the second, which essentially merges symbol A and B into a third symbol which is necessarily itself, but has been defined as conditionally itself.
Now we can conclude that if this sentence is true, then Santa Claus exists, then necessarily,
if Santa Claus does not exist, this sentence is false, but
if this sentence is false, we cannot conclusively make any statement about the existence of Santa Claus.
Even if you modified the above statement to
iff this sentence is true, then Santa Claus exists,
the above would still apply. If the sentence is false, then the iff condition is also false, so if the sentence is false, Santa Claus may or may not exist.