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It exists about as much as the number 2 or -4/7. Which is to say, not at all, in comparison to the physical and tangible existence of my laptop, the Eiffel Tower, or a squirrel. However, these numbers, as well as infinity, are used as abstract concepts within a system that categorizes, simplifies, explains, and predicts real world phenomena. For example, most of Physics that we know today was derived using methods of calculus. But without infinity, the methods of calculus make no sense, as limits cannot be defined on only a finite amount of terms. Now this is not to say that it is completely impossible to do meaningful Physics calculations without the assumption of infinite sets. However, there is not any meaningful work done in Physics (or any other field) in such an "ultrafinite" system of calculations except possibly as a novelty, due to the relative simplicity, elegance, and ease of use of systems that do contain infinite sets.
Rather than asking whether infinity has a real and tangible existence, one should ask whether mathematical models (such as the Peano axioms) which have an infinite amount as a concept of that system can be used as a model of real world phenomena and have that model produce useful results. Note that this distinction is not only for "infinity" but indeed any abstract concept whatsoever: "infinity", "two", "i (the imaginary number)", "triangle", "11-dimensional hypersphere", or "smooth surface". Note that even if a concept such as "11-dimensional hypersphere" does not necessarily have a direct physical analogue, the analyses of these objects still have use, as they can be used to find general truths which apply to similar, related objects.
