The Wish and The Word is a theoretical optimization (TO) exercise. That means it is not intended to be played at all, it is intended purely to experiment and play with the system’s rules. It also means that it is operating under the most-favorable-conceivable interpretations of RAW. And there is a... conceivable argument for each point in The Word. Many of them are dubious, even form a TO perspective, because it is really pushing the envelope.
Stacking multiple orange ioun stones
This is an extremely dubious interpretation of the rules, and wide consensus is against it. Certainly, I do not recommend ever allowing it in your game.
However, ultimately, this comes down to a question of what it means for something to be “the same source.” The rules prevent untyped bonuses from stacking when they come from the same source, but it’s not exactly clear what that means. Most people argue it’s when the source is the same sort of thing, usually hinging on whether or not it has the same name. Two orange ioun stones are two of the same thing, so their bonuses are from the same source.
The opposite argument, used by The Word, is that two orange ioun stones are two different physical objects, i.e. not the same source at all. This is rather dubious—after all, the explicit example of same-source failure to stack is the same spell cast twice. Using two of the same item is rather similar to casting the same spell twice.
For what it’s worth, “The Sage” Q&A column ruled against multiple orange ioun stones—as noted in our own Q&A on the subject of the FAQ, Wizards of the Coast’s “rules clarification” channels are notoriously unreliable, and that includes the Sage, but nonetheless, it’s worth knowing the answer exists.
Sublime chord caster level
Yes, the sublime chord can effectively “reset” one’s caster level in a number of arcane spellcasting classes, because it specifies that all of your arcane spells, both sublime chord spells and “the spells she gains from other arcane spellcasting classes” use the indicated formula, based on sublime chord level and level in one other arcane spellcasting class.
The big favorable interpretation here is that The Wish is counting other prestige classes that increase her caster level with wizard spells as counting towards her wizard level for the purposes of her global arcane caster level. That is, when sublime chord sets the caster level of all her arcane spells to \$L_\text{wizard} + L_\text{sublime chord}\$, she is treating \$L_\text{wizard}\$ as being equal to her wizard class level plus any prestige class levels that advance wizard spellcasting, and treating \$L_\text{sublime chord}\$ as being equal to not just her sublime chord level, but also any levels in prestige classes that are advancing sublime chord level.
This is doubtful, by RAW, because sublime chord refers to “levels in class” rather than any kind of “effective spellcasting level” or similar. That said, a most-favorable interpretation could include them, claiming that those prestige classes all say something like “these levels stack with those levels for the purposes of caster level.”
Unlike the multiple orange ioun stones, this one is a pretty common interpretation, even when not engaging in a TO exercise. However, and this is important, that goes out the window when you start abusing it—the sublime chord wording is a mess and is really counter-intuitive and confusing, so as long as no one’s being abusive, most people just let it work. When you start pulling shenanigans, though, people are usually going to get a lot more strict.
Ur-priest caster level
Ur-priest is similar to the sublime chord, and uses a similar mechanic to determine its caster level. Again, we are adding together the levels of other spellcasting classes to determine the sublime chord caster level. Unlike sublime chord, ur-priest does not set those class’s spells to the same value it uses itself. But also unlike sublime chord, ur-priest adds all those classes together, rather than just one of them.
The Word is using this to basically multiply her arcane caster level by the number of arcane classes she has, and then using half that product to determine her ur-priest caster level. That’s the whole gimmick.
The dubious part here is using “caster level” when ur-priest says “one-half his levels in other spellcasting classes.” Unlike the situation where prestige classes explicitly say they stack with class levels for the purpose of caster level, sublime chord setting the caster level does not say that. In fact, it actually says that it is those spells that use this calculation for caster level. Because that’s one of the dirty little secrets of D&D 3.5e—spells have caster levels, not spellcasters. Spellcasters have the caster level used by their spells (usually equal to their class level), it’s technically not “their caster level.”
However, we can make an abusive argument to get around this. This basically doesn’t fly unless you are trying really, really hard to convince yourself it does. Basically, we are going back to the wording on those prestige classes: they specify that those levels stack with the other class levels for the purpose of caster level. Since we are talking about caster level here, everything here is “for the purpose of caster level,” even if indirectly.
So now we are arguing that the sublime chord doesn’t just set the number used by other classes’ arcane spells, but actually setting those spells to use the entire stack of prestige class levels. As a result, sublime chord doesn’t set, say, the caster level of a bard spell to \$X\$, but sets it to \$\left(L_\text{wizard} + \sum{L_\text{prestige}}\right) + \left(L_\text{sublime chord} + \sum{L_\text{prestige}}\right)\$, treated “symbolically” if you will. And as a result, your “levels in other spellcasting classes” isn’t just the 1 level you’ve actually got—it’s the stack of levels that sublime chord set those spells to use, because those all stack with her class level “for the purpose of caster level.”
This argument is, basically, absurd. It requires, basically, wanting this result and twisting your head around any possible justification. But that can be what TO is.
Conclusion
If you buy those three arguments, they lead directly to The Word. So that is how you get to that point. It is what it is. I wouldn’t honestly worry about it too much.