How can someone reject a math result if everything has to be proved?
Although the possibility of different axioms is a concern, I think the major objection the author is speaking of is largely about constructivism (i.e. intuitionistic logic). There really is a big gap between rational numbers and real ones: with enough memory and time, a computer can represent any rational number and can do arithmetic on these numbers and compare them. This is not true for real numbers.
To be specific, but not too technical: let's start by agreeing that the rational numbers $\mathbb Q$ are a sensible concept - the only controversial bit of that involving infinite sets. A Dedekind cut is really just a function $f:\mathbb Q\rightarrow \{0,1\}$ such that (a) $f$ is surjective, (b) if $x<y$ and $f(y)=0$ then $f(x)=0$, and (c) for all $x$ such that $f(x)=0$ there exists a $y$ such that $x<y$ and $f(y)=0$.
Immediately we're into trouble with this definition - it is common that constructivists view a function $f:\mathbb Q\rightarrow\{0,1\}$ as some object or oracle that, given a rational number, yields either $0$ or $1$. So, I can ask about $f(0)$ or $f(1)$ or $f(1/2)$ and get answers - and maybe from these queries I could conclude $f$ was not a Dedekind cut (for instance, if $f(0)=1$ and $f(1)=0$). However, no matter how long I spend inquiring about $f$, I'm never going to even be able to verify that $f$ is a Dedekind cut. Even if I had two $f$ and $g$ that I knew to be Dedekind cuts, it would not be possible for me to, by asking for finitely many values, determine whether $f=g$ or not - and, in constructivism, there is no recourse to the law of the excluded middle, so we cannot say "either $f=g$ or it doesn't" and then have no path to discussing equality in the terms of "given two values, are they equal?"*.
The same trouble comes up when I try to add two cuts - if I had the Dedekind cut for $\sqrt{2}$ and the cut for $2-\sqrt{2}$ and wanted $g$ to be the Dedekind cut of the sum, I would never, by querying the given cuts, be able to determine $g(2)$ - I would never find two elements of the lower cut of the summands that added to at least $2$ nor two elements of the upper cut of the summands that added to no more than $2$.
There are some constructive ways around this obstacle - you can certainly say "real numbers are these functions alongside proofs that they are Dedekind cuts" and then you can define what a proof that $x<y$ or $x=y$ or $x=y+z$ looks like - and even then prove some theorems, but you never get to the typical axiomatizations where you get to say "an ordered ring is a set $S$ alongside functions $+,\times :S\times S \rightarrow S$ and $<:S\times S \rightarrow \{0,1\}$ such that..." because you can't define these functions constructively on $\mathbb R$.
(*To be more concrete - type theory discusses equality in the sense of "a proof that two functions $f,g$ are equal is a function that, for each input $x$, gives a proof that $f(x)=g(x)$" - and the fact that we can't figure this out by querying doesn't mean that we can't show specific functions to be equal by other means. However, it's a huge leap to go from "I can compare two rational numbers" - which is to say, I can always produce, from two rational numbers, a proof of equality or inequality - to "a proof that two real numbers is equal consists of..." understanding that the latter definition does not let us always produce a proof of equality or inequality for any pair of real numbers)
This is somewhat surprising if you're not used to this. But of course you're free to reject whatever mathematical statements you dislike. The real question is what else you are forced to reject with it, and what would remain of the mathematics that you know and love otherwise.
The onus is on you, as someone who decided that "everyone else is wrong", to convince people that your idea is better, and to get people to take interest in what and how to transfer mathematics from "the realm of error" into "the world of truth". That is, until someone will come in and reject your ideas, etc.
For example, Lebesgue is well-known as someone who rejected the axiom of choice. For him, the existence of non-measurable sets was unthinkable, so he was forced to reject the axiom of choice, and many other theorems that would contradict that.
Another example is in Kronecker who rejected the idea that infinite sets exist, this means that for Kronecker the axiom of infinity would be false. That implies that we want to work, in some sense, with some a second-order theory over the natural numbers, we can get some analysis done, and everything beyond that would be "a fiction".
Many people would reject large cardinal axioms, those are easily misunderstood and mistrusted outside of set theory (although often ignored just as well). But without inaccessible cardinals, there are not Grothendieck universes; without measurable cardinals there are some accessible categories which are not well-copowered. Even some set theorists reject large cardinal axioms such as Reinhardt and Berkeley cardinals, since they imply the negation of the axiom of choice, which (unlike Lebesgue) most set theorists readily accept as "obvious truth".
What is true, is that there is an implicit theory underlying mathematics, which lets us develop "most of working mathematics" without having to worry about foundations. But this theory is not without its controversies. It includes infinite sets, the axiom of choice, the law of excluded middle, and more. Sometimes it is just interesting to see what part actually depends on these axioms, and sometimes people outright feel that something is wrong with the axioms.
If you are more inclined to use computer assistance in your work (e.g. proof verification software), you might be more inclined to take a different foundation which is easier to understand from your proof assistant's point of view. This may be something that rejects the LEM, for example, or otherwise is incongruous with what "most people" would call "every day mathematics".
Remember that the book you are reading is on axiomatic set theory. Any time you do pure mathematics, you have to start with axioms. You can't prove them, you just specify them. And then you use them to prove other things.
The famous example of this is the parallel postulate. People were surprised when it was realized that you could have a perfectly consistent geometry where there was an infinite number of lines through a point and parallel to another line (not on the point).
In set theory, the axiom of choice plays a similar role. You cannot prove it from the other axioms, but yet it feels more like a theorem than a lot of the other axioms. Most people find it intuitively true, but some do not.
The different "schools" are people with different opinions about which sets of axioms you should use. They are not mainstream, but unlike fringe groups in other fields, nobody doubts the validity of the math that they do. The question "If you reject the axiom of choice, what can you prove?" is perfectly legitimate.