What does it mean when we say a mathematical object exists?

Arguably, your question cannot be answered in a satisfying way (unless you're a formalist).

Ultimately most mathematicians don't spend too much time thinking about ontology - a sort of "naive Platonism" may be adopted, although when pressed I think we generally retreat from that stance - but I think the "standard" meaning is simply, "The existence of such an object is provable from the axioms of mathematics," and "the axioms of mathematics" is generally understood as referring to ZFC. So, e.g., when we say "We can prove that an object with property $P$ exists," what we mean is "ZFC proves '$\exists xP(x)$.'" This is a completely formalist approach; in particular, it renders a question like

does that mean that there is some real number out there, but uncatchable somehow by the nature of its existence?

irrelevant, since there is no "out there" being referred to. It is also completely unambiguous (up to a choice of how we express the relevant mathematical statement in the language of set theory). Of course, if you give me a precise notion of "construct" (or "catch") then the question "Why does (or does?) ZFC prove the existence of a non-constructable (deliberately misspelled for clarity) object?" is something we can address, but now it's not really about the nature of mathematical existence but rather the nature of ZFC as a theory.

This response does ultimately just push the question back to why we privilege ZFC (and classical logic), and dodging this question (and any other formalist answer) grates against "realist" sensibilities. At the end of the day, the nature of existence is a philosophical, rather than mathematical, question; to my mind one of the main values of formalism is that it provides us with a language for doing mathematics which bridges philosophical differences. E.g. a large-cardinal-Platonist, an intuitionist, and an ultrafinitist will all agree with the statement "ZFC proves that there is an undetermined infinite game on $\omega$," regardless of their opinions on the statement "There exists an undetermined infinite game on $\omega$."