We like stable homotopy groups because of the Freudenthal suspension theorem which tells us that homotopy groups stabilise after many suspensions.

The basic idea seems to be something like a tensor-hom adjunction. We have the loop spaces which are like $S^1 \rightarrow X$ and the suspension which is like $S^1 \wedge X$. The theory begins by considering the tensor-hom-adjunction between these objects as fundamental. So curry stuff around to write things as (S^1, A) -> B and A -> (S^1 -> B), which is Suspension(A) -> B and A -> Loop(B). This gives us the adjunction between suspension and looping.

• We then try to ask: how can one invert the suspension formally? One tries to do some sort of formal nonsense, by declaring that maps between $\Sigma^{-n}X$and $\Sigma^{-m} Y$ , but this doesn't work due to some sort of grading issue.
• Instead, one repaces a single object $X$ with a family of objects $\{ X_i \}$called as the spectrum. Then, we can invert the suspension by trying to invert maps between objects of the same index.

§ References

• Homotopy Extension
• CW complexes
• Stable homotopy theory 1