This is one of the motivations given in Adams’ book, which you should read… modulo the part on smash products.
Given a finite CW-complex, \(X\), I can embed it into a large sphere \(S^n\) in a nice way so that the complement deformation retracts onto a finite CW-complex, \(Y\). This is called the Spanier-Whitehead dual of \(X\).
The thing is, \(Y\) is not determined up to homotopy by \(X\). However, the stable homotopy type of \(Y\) is determined by \(X\), independent of the choice of embedding or sphere. This basically follows from Alexander duality. So if we want to make arguments exploiting duality, it’s best to work in the stable category.
