What if any important results in real analysis make use of the notion of an "undefinable" real number? (Whatever "important" may mean to the reader.) Or is it used more in the philosophy of mathematics?
The notion is important in mathematical logic and model theory, but not in classical mathematics, including real analysis as traditionally understood. Definable predicates are generally important in the theory of formal systems because they show how expressive they are, for example Tarski's theorem on the undefinability of truth states that in a consistent formal system that includes basic arithmetic the truth predicate is undefinable (this is closely related to Gödel incompleteness).
Undefinable real numbers are the numbers that can not be uniquely described by definable predicates, so the notion is useful in studying the language of analysis rather than its objects, which is the main subject of the classical analysis. Still, undefinable numbers can help express certain facts about the higher reaches of the universe of sets. My favorite is the undefinable real number 0# discovered by Silver in 1966. Its existence is unprovable in the standard set theory (ZFC), but has a decisive effect on the structure of the universe of sets. If 0# does not exist then the universe of sets looks a lot like Gödel's constructible universe, which is well-behaved and nicely describable, if it does then not only is it non-constructible but generally wild, in particular every uncountable cardinal of the universe is "ineffable" within its constructible part.
Running off the idea that there are undefinable numbers because there are countably infinite definable real numbers and uncountably infinite real numbers, one result that could be considered "important" is that no continuous chaotic system can be perfectly modeled by a Turing machine. Thus, if a truly chaotic dynamic system exists, it could not be part of a simulation of the universe that is running on a Turing machine.
To paraphrase Joel Hamkins answer (pointed out by user4894) on the notion of undefinable reals.
The naive account of undefinability points out there is only a countable number of ways we can describe a number, but there is an uncountable number of reals, hence there must be reals that we can't describe; however the notion of definability is problematical: since the real line can be well ordered (in ZFC) we can ask for the least undefinable positive real; but this defines it, and also by construction it is undefinable and so this has led us straight to a contradiction.
At least part of the problem here is the nature of the logical language we use with ZFC.
The point is that the concept of definability is a second-order concept, that only makes sense from an outside-the-universe perspective. Tarski's theorem on the non-definability of truth shows that there is no first-order definition that allows us a uniform treatment of saying that a particular particular formula is true at a point and only at that point.
Hence the notion of definability is important in suggesting that the first order logic isn't sufficient.
I would just like to point out, and I'm doing so in an answer and not in a comment because I do not have sufficient reputation, that the standard "least undefinable number" strategy that @Mozibur Ullah mentions does not work for the real numbers.
In order for this to work, the structure needs to be not just ordered, but well-ordered. JDH (mentioned in Mozibur's answer) talks about ordinals, which are; the real numbers are not.
Because the real numbers are not well-ordered (under the standard ordering, and without the Axiom of Choice, it is possible they have no well-ordering), it is possible that for any real number, there is an undefinable real number less than it, so there is no least undefinable real number.