Does mathematics require axioms?
Is it true that with modern mathematics it is becoming less important for an axiom to be self-evident?
Yes and no.
Yes
in the sense that we now realize that all proofs, in the end, come down to the axioms and logical deduction rules that were assumed in writing the proof. For every statement, there are systems in which the statement is provable, including specifically the systems that assume the statement as an axiom. Thus no statement is "unprovable" in the broadest sense - it can only be unprovable relative to a specific set of axioms.
When we look at things in complete generality, in this way, there is no reason to think that the "axioms" for every system will be self-evident. There has been a parallel shift in the study of logic away from the traditional viewpoint that there should be a single "correct" logic, towards the modern viewpoint that there are multiple logics which, though incompatible, are each of interest in certain situations.
No
in the sense that mathematicians spend their time where it interests them, and few people are interested in studying systems which they feel have implausible or meaningless axioms. Thus some motivation is needed to interest others. The fact that an axiom seems self-evident is one form that motivation can take.
In the case of ZFC, there is a well-known argument that purports to show how the axioms are, in fact, self evident (with the exception of the axiom of replacement), by showing that the axioms all hold in a pre-formal conception of the cumulative hierarchy. This argument is presented, for example, in the article by Shoenfield in the Handbook of Mathematical Logic.
Another in-depth analysis of the state of axiomatics in contemporary foundations of mathematics is "Does Mathematics Need New Axioms?" by Solomon Feferman, Harvey M. Friedman, Penelope Maddy and John R. Steel, Bulletin of Symbolic Logic, 2000.
Disclaimer: I didn't read the entire original quote in details, the question had since been edited and the quote was shortened. My answer is based on the title, the introduction, and a few paragraphs from the [original] quote.
Mathematics, modern mathematics focuses a lot of resources on rigor. After several millenniums where mathematics was based on intuition, and that got some results, we reached a point where rigor was needed.
Once rigor is needed one cannot just "do things". One has to obey a particular set of rules which define what constitutes as a legitimate proof. True, we don't write all proof in a fully rigorous way, and we do make mistakes from time to time due to neglecting the details.
However we need a rigid framework which tells us what is rigor. Axioms are the direct result of this framework, because axioms are really just assumptions that we are not going to argue with (for the time being anyway). It's a word which we use to distinguish some assumptions from other assumptions, and thus giving them some status of "assumptions we do not wish to change very often".
I should add two points, as well.
I am not living in a mathematical world. The last I checked I had arms and legs, and not mathematical objects. I ate dinner and not some derived functor. And I am using a computer to write this answer. All these things are not mathematical objects, these are physical objects.
Seeing how I am not living in the mathematical world, but rather in the physical world, I see no need whatsoever to insist that mathematics will describe the world I am in. I prefer to talk about mathematics in a framework where I have rules which help me decide whether or not something is a reasonable deduction or not.
Of course, if I were to discuss how many keyboards I have on my desk, or how many speakers are attached to my computer right now -- then of course I wouldn't have any problem in dropping rigor. But unfortunately a lot of the things in modern mathematics deal with infinite and very general objects. These objects defy all intuition and when not working rigorously mistakes pop up more often then they should, as history taught us.
So one has to decide: either do mathematics about the objects on my desk, or in my kitchen cabinets; or stick to rigor and axioms. I think that the latter is a better choice.
I spoke with more than one Ph.D. student in computer science that did their M.Sc. in mathematics (and some folks that only study a part of their undergrad in mathematics, and the rest in computer science), and everyone agreed on one thing: computer science lacks the definition of proof and rigor, and it gets really difficult to follow some results.
For example, one of them told me he listened to a series of lectures by someone who has a world renowned expertise in a particular topic, and that person made a horrible mistake in the proof of a most trivial lemma. Of course the lemma was correct (and that friend of mine sat to write a proof down), but can we really allow negligence like that? In computer science a lot of the results are later applied into code and put into tests. Of course that doesn't prove their correctness, but it gives a "good enough" feel to it.
How are we, in mathematics, supposed to test our proofs about intangible objects? When we write an inductive argument. How are we even supposed to begin testing it? Here is an example: all the decimal expansions of integers are shorter than $2000^{1000}$ decimal digits. I defy someone to write an integer which is larger than $10^{2000^{1000}}$ explicitly. It can't be done in the physical world! Does that mean this preposterous claim is correct? No, it does not. Why? Because our intuition about integers tells us that they are infinite, and that all of them have decimal expansions. It would be absurd to assume otherwise.
It is important to realize that axioms are not just the axioms of logic and $\sf ZFC$. Axioms are all around us. These are the definitions of mathematical objects. We have axioms of a topological space, and axioms for a category and axioms of groups, semigroups and cohomologies.
To ignore that fact is to bury your head in the sand and insist that axioms are only for logicians and set theorists.
It seems that many people regard the author's view as naive or un(der)informed. I disagree.
There is a well-known phrase attributed to Kronecker (presumably originally stated in German, and perhaps I am slightly misquoting the English translation as well) that "God created the natural numbers, and all else is the work of man". This is (in my view) an essentially anti-axiomatic declaration, which aligns fairly closely with the point of view in the essay under consideration, namely that mathematics is the investigation of certain "god-given" objects, such as the natural numbers, or the Lie group $G_2$ (to take an example from the essay).
This view is partly Platonist (in the sense that that term is generally used in these sorts of discussions, referring to a belief in a non-formal mathematical reality) and partly constructivist. It is one that I'm personally sympathetic to, and I don't think I'm alone in that. I regard ZFC as a convenient framework for doing mathematics in, but not as the actual basis underlying the mathematics I do; the natural numbers and the investigation of their properties are (in my view) much more fundamental than ZFC or other axiomatic systems that might encode them --- and the same goes for $G_2$ (again in my view)!
My view might be a minority one among working mathematicians (I don't really know), but I know that I'm not the only one who holds it. I also know others who genuinely believe that everything they do rests on ZFC, and that this is of crucial importance.
Another thing: it is often said that even though many mathematicians don't explicitly invoke the axioms of ZFC in their work, they are implicitly resting on those foundations. Personally, I don't find this convincing; I think it is often the case that those who do believe that everything necessarily rests on ZFC find it easy to construe what others are doing as (implicitly) resting on those foundations. But those who don't believe this also won't accept the claims that their work implicitly relies on those foundations.
Just to be clear, by the way: my comments here are not meant to apply to things like theorems in group theory, or commutative algebra, or Lie theory, where one derives consequences from the axioms that a structure satisfies (although they might apply in certain contexts where set-theoretic issues potentially intervene); obviously there axioms play a role, although, as the author writes, in these contexts axioms might be better construed as definitions. Rather, they apply to the basic objects of mathematics like the natural numbers, Diophantine equations, and so on.
It also seems worth mentioning something here which I also made a comment about on another answer:
It doesn't seem to currently be known whether FLT is proved in PA, or only in some more sophisticated axiomization of the natural numbers. On the other hand, there is no doubt among number theorists that the proof is correct. How is such a situation possible? In my view, it's because people ultimately verify the proof not by checking that it is consistent with some specified list of axioms, but by checking that it accords with their basic intuition of the situation, an intiuition which exists prior to any axiomization.
In the end, it will presumably be possible to isolate precisely those properties of the natural numbers that are used in the proof, whether it is the axioms of PA or something stronger, but my point is that the proof is known to be correct although what precise properties of $\mathbb N$ are being used is not yet known! This is because we can argue about $\mathbb N$ based on our intrinsic understanding of it, without having to encode all the aspects of that understanding that we use in precise axiomatic form.