Even though I'm personally not much of a realist, I feel like it's worth explaining why it's so popular when studying foundations of mathematics, because outdated is selling it short and it's actually a fairly nuanced position. The basic question is are mathematicians fundamentally discovering mathematics (Platonism) or creating mathematics (non-Platonism)?
The first step is that it seems like natural numbers are "real" and "tangible" in some sense. So for example, even though we can have modular arithmetic that would say maybe 2 + 5 = 2 under addition mod 5, it clearly feels "artificial" compared to 2 + 5 = 7. More concretely there seems to be some universal idea that seems to underly the fact that if you have three stones and add another stone you get four stones, or if you have three bottles and add another bottle you get four bottles. "three", "one", and "four" might not exist in the same as "bottle" or "stone" exist, but clearly they seem to exist and are "true" in some sense (it is clearly incorrect to say if you have three bottles and add another bottle you still have three bottles).
Indeed, to even talk about mathematics it seems like you need some intuitive grasp of natural numbers that precedes mathematical axioms, or at least be able to understand at a deep level that if you have n things and you add another thing, then you have n + 1 things, and that any axioms that violate this are more "artificial" than axioms which don't violate this. More generally, if you have axioms that create results which violate the rules of "normal" natural numbers, it seems reasonable to say that those axioms are "artificial" and not "real" in some sense. And so one interpretation of mathematicians' jobs, at least when tightly scoped to natural numbers, is to find those axioms of the natural numbers that reflect our natural numbers really work in "our universe" as opposed to a hypothetical other universe. And that approach seems to presuppose that natural numbers exist, at least in our universe, in some objective some that we are "discovering" with our axioms, rather than "creating" ex nihilo.
Indeed being able to say the term "normal natural numbers" and have even lay, non-mathematicians understand what that means (as opposed to "weird natural numbers" such as modular arithmetic) seems to suggest a certain objectiveness to the natural numbers.
Or to put it another way, the abstract notion of "counting" seems to exist objectively and outside of our formal mathematics theories, and our formal theory of natural number is really trying to describe "counting" rather than create the notion of "counting" from scratch.
Now if you accept that "counting" as an abstract idea exists in some sense that makes it possible to say either "yes that is an accurate description of counting" or "no that is not an accurate description of counting," then Godel's incompleteness theorems become quite interesting, namely that every useful theory of the natural numbers will have additional axioms that are independent of the ones already included in that theory. The non-realist looks at the incompleteness theorems and says, "Well yeah those new axioms just create different 'natural numbers' no biggie." The realist says, "That seems totally at odds with the fact that even though hypothetically you could have multiple notions of 'natural number,' there is only one notion of 'natural number' that holds true in our current universe and agrees with our similarly abstract notion of 'counting' and not a hypothetical other universe!"
And if you accept that for the natural numbers... well a lot of things have ramifications for the natural numbers. One classic example, as mentioned elsewhere in this HN discussion, is the busy beaver function, a function whose input is the number of instructions in a given Turing machine and whose output is the maximum number of steps the machine could possibly take for any input before it must be non-terminating on that input (basically an end run around the halting problem).
Any set of axioms (including ZFC) that includes the notion of arithmetic makes a statement on what it think a finite number of busy beaver values are (and only a finite number, again the halting problem prevents us from knowing more). And yet it seems like every value of the busy beaver function has some objective truth value in our universe! I can just run the Turing machine and find out! And that seems like an objective yardstick that we can use to determine whether a given axiom system is "true" or not in our universe (granted it's not one of very much utility because you are waiting to see if a machine runs forever, but still something that seems to have obvious physical ramifications).