I'm not sure I entirely got the point (I find it weird to say that the variable can be eliminated in natural language, because 1. you don't explicitly use variables in natural language anyway and 2. "there is some fox..." is not a variable, so I find that wording misleading), but I think it's important for you to have the the basic definitions in mind (sorry for the rudimentary formatting and sloppy wording):
- [[P(x)]]^g is true iff g(x) is in the denotation of P
- [[ExP(x)]]^g is true iff there is an x-alternative g' of g such that g'(x) is in the denotation of P
where g is an assignment function mapping from variables to individuals in the domain, and an x-alternative of g is a variable assignment that is just like g except for possibly the value it assigns to x.
This means that "fox(y)" without a quantifier is true only if the current variable assignment happens to assign to y an individual which is in the denotation of the predicate "fox" in the current model, i.e. if THAT VERY y stands for an individual which is a fox (and is disliked by x). Since we stick to one particular variable assignment and don't quantify over other possible variable assingments, the free variable y behaves just like an individual constant.
On the other hand, with the existential quantifier, the formlula is true if there is SOME individual in the domain which is able to fulfill the fox predicate, i.e. if there is some way of mapping the variable y to some element of the actual world which is also a fox in the actual world, i.e. if there is at least one fox (and that fox is disliked by x).
What I think the paragprah wants to say is that you don't need to explicitly spell out the quantifier in natural language (i.e. not necessarily "There is some y that is a fox and x dislikes it" but rather "x dislikes a fox"), while you can't eliminate the constant (you need to say "x dislikes THE FOX", where "the fox" is referring to the denotation of g(y), the sentence behaves just like "x dislikes John", where "John" also refers to an individual constant).
This is also why in the first example (without the quantifier), you have the definite article "the", because it refers to a single individual, while the second example (with the quantifier) ranges over possibly several individuals where you need to use the indefinte article "a" (which obvisoulsy in natural language usually has the meaning of "only one", which with the existential quantifier would have to be stated explicitly though; and of course, the set of foxes disliked by x might happen to be a singleton, but the logical type is still a different one).
Regarding your comment:
"the fox named y" doesn't make sense anyway. y is not the name of the fox, but a variable we use in the formalisation of the sentence in order to represent the logical entities involved and map placeholders for individuals to their "real-life counterparts" of our domain. "y" behaves somewhat like a name in the formalisation of a sentence in referring to a certain individual, but it is certainly not the name of the actual fox in the extension the sentence the notes. You need to be carful with mixing up different language levels.
I hope this at least in some aspect an answer to your question; as I said, I'm not sure I got the point of it without more context and am little confused by the wording.
Edit: My answer is closely related to that in your other thread: https://linguistics.stackexchange.com/a/17744/13238