No philosophy of math that I've encountered seems to say much of meaning about mathematics as I've actually done and understood it, and most of it seems ridiculous to me. So let me set down my own meaningless and ridiculous understanding, and say a bit in particular about why mathematics as it's actually done and experienced is a type of narrative. I'm going to have to use a lot of attack quotes in what follows; attack quotes will denote a word that applies in some senses and not others. (That's all words, really, but...)
I perceive mathematics in two aspects: it has a meaningless/formalist face and an empirical or quasi-empirical face.
The formalist aspect of mathematics is the axiomatic face: like choosing rules for a game. Imagine a set of rules, when fixed, instantly spinning out a universe of consequences. Most of the actual work of math is exploring that universe. Think of the rules of chess as 'creating'--more accurately allowing the creation of, opening a space for--every specific game of chess ever played, and astronomically more games that never have been played. Think of the axioms of Euclid's Elements--each one agonized over as an understanding of reality, but once fixed and joined together freed of that relation and 'creating' (allowing, opening up) the conceptual world of geometry.
Any such universe, any such web of consequences, is 'objective' inasmuch as it's unyielding to any influence but itself. But I can churn out these universes forever, by picking different axioms (rules, games): exploring some will be trivially boring, others will be complex and interesting, and still others will be overwhelmingly, impossibly difficult. As such 'universes' can be 'created' in unlimited profusion, and discarded if I find them dull or annoying, it's odd to claim that they 'exist'. Sort of like creating a hologram and then peering through it to see the 'universe' it depicts. The image isn't 'real' in most senses that we understand that word, but it's real-ish as an emergent phenomenon defined by the structure of the film. (You made the film itself, so that's real, but arbitrary.) Nevertheless, you might see something interesting and unexpected in the image. Might see a whole ghostly world that you could never have conceptualized yourself. Think of the Mandelbrot set, defined by a few simple rules and the equation zn+1 = zn2 + c, and the infinitely complex image it generates.
The quasi-empirical face of mathematics is where we can see mathematics as narrative--the exploration of these ephemeral universes is something that we experience over time, and encoding that exploration in a coherent temporal/symbolic/conceptual way is precisely the art of narrative. Every chess game is a story, with characters, a journey, conflict, and resolution, and so is every theorem. (Most of them are very badly told stories, but that's Sturgeon's Law.) Choosing axioms (rules, games, universes) is a tiny fraction of what mathematicians actually do. Most of a mathematician's work consists of journeying into these ghost-worlds, and telling stories about the expedition.
tl;dr: Most philosophy of mathematics overloads the word 'exists' until it becomes useless.