I learned recently that there are mathematical objects that can be proven to exist, but also that can be proven to be impossible to "construct". For example see this answer on MSE:

Does the existence of a mathematical object imply that it is possible to construct the object?

Now my question is then, what does existence of an object really mean, if it is impossible to "find" it? What does it mean when we say that a mathematical object exists?

Because the abstract nature of this question, and to make sure I understand myself what I'm asking, here is a more specific example.

Say that if have shown that there exists a real number $x$ that satisfies some property. I always assumed that this means that it possible to find this number $x$ in the set of real numbers. It may be hard to describe this number, but I would assume it is at least possible, to construct a number $x$ that is provable a real number that satisfies the property.

But if that does not have to be the case, if it is impossible to find the number $x$ that satisfies this property, what does this then mean? I just can't get my mind around this. Does that mean that there is some real number out there, but uncatchable somehow by the nature of its existence?