So I'm no philosopher of math, but the intuitive way I think of it is that negative numbers are like borrowing a place holder.
Think about how electrical charge and currents work at the physical level. There's electrons, which have negative charge, or holes, which have positive charge. Holes are just an empty place an electron can go, rather than an extant particle.
Similarly, when we think of negative numbers in relation to counting numbers, we're just using a notational trick to keep track of a place holder or hole, a spot where a unary count can potentially go later to cancel it out.
There's a pretty fascinating book named Quantum Computing Since Democritus by Scott Aaronson, one of the leaders in that field. The idea of the book is "Could the ancient greeks have discovered quantum mechanics?"
Much of the higher level math in the book is past my familiarity, but the central theme is clear and intuitive: if you take ordinary probabilities, generalize them to allow negative probabilities, and then generalize those to allow complex numbers, out pops quantum mechanics quite naturally.
Why do these two generalization steps make sense? What the heck is a negative probability of an event? It's exactly just notational borrowing in the same sense as above. Why generalize to complex numbers? That's more tricky, but I think of it from two directions: 1. It allows you to model partial constructive and destructive interference of probabilities, rather than just simple union or intersection. 2. Complex numbers are algebraically closed, while more simple numbers are not. So it feels natural that our ultimate number system to model nature would need to extend all this way.
I realize some math heavy folks would find my way of thinking of this a bit hand wavy, but it really has helped me cut through the confusion and mystery. It also fits in very well with a bayesian perspective on probabilities.
This, the "fields are what's real" perspective on physics, and bayesian epistemology in general have greatly simplified the way I think of these big idea question topics.