What does it mean for a set to exist?

There is no definition of what a set is. The reason is simple: it is hard to imagine anything more fundamental than a set that you could use in order to say what a set is. And if there were such a thing, let's call it a tet, then one would ask "but what exactly is a tet? can we define it in terms of something more fundamental, perhaps in terms of a uet or a vet or a wet or xet or a zet?". This never ends.

So, what we do is we take one of two approaches. The naive approach resorts to trying to brush the vagueness away by saying things like "a set is a collection of things where order is not important, and repetition is not possible". While not incorrect, what the hell is a collection then? Well, in the naive approach we don't care and just assume we all share the same intuition as to what sets are. This works up to a certain degree.

The other approach is the axiomatic one. Instead of saying what sets are, we say what we can do with them and which laws govern them. There are many different axiomatizations of set theory, none of which is particularly simple (e.g., requires a bit of effort to understand). Fundamental results in axiomatic set theory include such results as "it is impossible to prove the consistency of set theory", essentially saying you can't be sure that a universe of sets as we pretend to exist really exists. Other results show that certain fundamental axioms are independent of others, showing that what one may consider 'obvious' about her idea of sets need not be so obvious for others.

Sets are objects in the universe of [pure] set theory.

Informally, sets are formalization of the idea of a collection of mathematical objects.

As for the existence, existence [of a set] in the pure mathematical sense means that in a mathematical universe there is a set with particular properties. So the real question is what is a mathematical universe? It is a collection of mathematical objects which obey particular laws which we call axioms.

So does a mathematical universe exist? This is a borderline theological question. It requires belief and personal conviction. But one can do mathematics in either case. It is possible to think of mathematics as a formal manipulation of strings; or to make-believe that these objects exists, for the sake of argument; or to believe that there is some idealized universe in which the mathematical objects reside; or even believe that our universe is a mathematical one.

Each approach has its merits and its downsides, and I do not wish to discuss those. One should follow their heart and mind, and decide for themselves.

Let me add a bit on the difficulty that people have with this concept when they meet it.

What is a real number? Well, it is the limit of rational numbers. What are rational numbers? Fractions, ratios between integers. What are integers? Integers are all the things you can have by adding or subtracting $1$ from itself indefinitely.

What is $1$? Well, I have one head. That's one.

That was relatively easy, because we have a firm grasp about what is "one thing", and what is "addition of two things", so we can understand the natural numbers naturally, and then we can construct other systems. Furthermore in modern times we all hear about physics about how the time is a continuum, and it is engraved into the common knowledge of mankind. This is why many, if not all, people think about the time line when they think about real numbers.

In fact once I sat with a few folks that were engineering students, and they were told that a four-dimensional cube is best thought as a regular cube moving through time (and collecting its "trail" somehow). I don't think any of my teachers would have dared to tell students something like that.

But sets, what are sets? We don't have sets in real life, at least not in the mathematical sense, and even the collections we do have in real life are not often identified with sets. This causes a mental strain when we try to give sets some physical manifestation.

If $\pi$ is the length of half a circle with radius $1$; and if $\frac23$ is the number of "sons/children" my parents have; and if $1$ is the loneliest number... then what is a "set"?

If we consider sets as collections without order or repetition, then we can think of sets as families, or classrooms, or nations, or the clowder of alley cats living around the local dumpster. These are collections of objects, without repetition and without a particular order.

But here's the kicker. We are used to mathematical objects signifying quantity, this much is obvious from how the common person often mistakes mathematics to a bunch of equations we "solve for $x$". If that is the case, what quantity do we measure with sets?

The answer may shock you, we don't measure quantities with sets, and not every mathematical object is a number or signifies quantity. Rather mathematical objects represent structure, and quantity can be thought of as a form of structure. In that case, sets represent the minimal structure possible, on which we add more. But let me stop here, this post is long enough as it is.