Those mathematical axioms which are taken from experience work so well because reality works so well. Those mathematical axioms which are free and often counterfactual inventions don't work well. A prominent example is transfinite set theory which has not the least practical application. For debunked examples see: "Applications" of set theory in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 124ff.
"How is it possible that mathematics, which is a product of human thinking independent of all experience, fits reality in such an excellent way?" [A. Einstein: "Geometrie und Erfahrung", Festvortrag, Berlin (1921), reprinted in A. Einstein: "Mein Weltbild", C. Seelig (ed.), Ullstein, Frankfurt (1966) p. 119]
Without mental images from sensory impressions and experience thinking is impossible. Without reality (which includes the apparatus required for thinking as well as the objects of thinking – we never think of an abstractum "number 3" but always of three things or the written 3 or the spoken word or any materialization which could have supplied the abstraction) mathematics could not have evolved like a universe could not have evolved without energy and mass. Therefore real mathematics agrees with reality in the excellent way it does.
Einstein answers his question in a relativizing way: "In so far the theorems of mathematics concern reality they are not certain, and in so far as they are certain they do not concern reality." [A. Einstein: "Geometrie und Erfahrung", Festvortrag, Berlin (1921), reprinted in A. Einstein: "Mein Weltbild", C. Seelig (ed.), Ullstein, Frankfurt (1966) p. 119f]
He states a contraposition (R ==> ¬C) <==> (C ==> ¬R). Both statements are equivalent. Both statements are false. To contradict them a counterexample is sufficient. A theorem of mathematics is the law of commutation of addition of natural numbers a + b = b + a. It can be proven in every case in the reality of a wallet with two pockets.