Spectra arise in the wild when studying a collection of functors (indexed by $n$) $$F_n: \{\text{CW-complexes}\} \to \text{AbGrp}$$

Sometimes, there is an object $E_n$ in the source category (in this case, CW-complexes) of $F_n$ such that $F_n(X) \simeq [X, E_n]$. When $F_n(-)$ is a cohomology functor, we call these $E_n$ “spectra.”

Spectra also fill the need for negative dimensional spheres — the need for a category where suspension has an inverse and not just an adjoint!

*If you want a fantastic introduction to the stable category of spectra, and the context of various topological theorems calling for a definition of negative dimensional spheres, this might help.*