I'm not sure how technical you want to get, but in formal set theory, this is usually done by an axiom--- that is, one of the assumptions you make (at the beginning, without any attempt at proving it or verifying it) is the assertion that there is an infinite set. The axiom that does this in the most common formalization of set theory is called the "axiom of infinity" and is included in the link below.
This may seem like "cheating", but it's worth pointing out that for most purposes you really only need to assume the existence of _one_ infinite set, and then you can build any other infinite sets you might need for most standard mathematics out of it. If you look in a book on number systems you will see, for example, how to define the integers, rational numbers, real numbers, complex numbers, and other things (e.g. the set of all functions from the real numbers to the real numbers) in terms of the set of natural numbers and other fundamental operations of set theory. So it isn't really necessary to separately assume the existence of all of the infinite sets that math wants to talk about. Just one is enough.
To flesh that out a little more, once you have the natural numbers, you can get "smaller" infinite sets from "subset axioms" (sometimes called comprehension axioms, or specification axioms) which assert that if you know a set S, and you have a well-formed property that an element of S might or might not have, then there is a set consisting of all elements of S that have that property. And you can get "larger" infinite sets from other axioms of set theory, which given a known set or sets, produce "bigger" sets (like the power set axiom--- asserting that if S is a set, there is a set whose elements are the subsets of S--- or the pairing axiom, or union axioms, which assert that if you know A and B are sets, there is a set of ordered pairs (a,b) of an element of A and an element of B, and a set whose elements are precisely those of A together with those of B). So any infinite set you might want to construct, would simply come from various axioms of set theory applied in various sequences to the single infinite set whose existence you assumed.
I should point out that if you are working informally--- ie, if "sets" are just language for discussing collections of mathematical objects--- there is really no technical issue at all: if you feel that you informally "know" infinite things (like the natural numbers) and can sensibly talk about them (e.g. to decide whether or not something is, or is not, in some infinite collection), then of course infinite sets "exist" for you, because you are capable of sensibly talking about them, and that's all you mean by the existence of a set in the first place.
I should also point out that it is equally valid to adopt the position that infinite sets simply do not exist. (Or, what amounts to the same thing, but perhaps sounds less absolutist--- to decline to talk about infinite sets.) This stance is often called "ultrafinitism" and some mathematicians do adhere to it, or at least adopt it from time to time. If you are interested in mathematical objects only to the extent that they can be explicitly computed and calculated, it is not that bizarre to argue that there is no use in assertions about infinite things. (Anything you might say about natural numbers for example, probably has the same practical content if you replace it with an assertion about numbers less than some fixed gigantic number. If N is the number of elementary particles in the universe, for example, there is a real sense in which numbers larger than N are somehow "fictitious" in a way that small numbers like 5 and 10 are not.)